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THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


GIFT  OF 
UC  Library 


No.  7. 


PAPERS 


PRACTICAL  ENGINEERING, 


PUBLISHED  BY  THE  ENGINEER  DEPARTMENT, 


FOR    THE    USE    OF   THE 


OFFICERS    OF    THE    UNITED    STATES    CORPS    OF    ENGINEERS. 


PAPKRS    ON    PRACTICAL    ENGINEERING. 


No.  7. 


TREATISE 


VARIOUS   ELEMENTS   OF    STABILITY 


WELL-PPiOPOPJIONED  AUCH. 


gumdrous  iablofi  of  i\u  tllimixk  and  %tiu\  Slxml 


BY 

CAPTAIN    D.  P.    WOODBURY, 

U.    S.    CORPS    OF   ENGINEERS. 

JOHIV  S.  Pi^ELL         ..•^^■"' 


KEW  YOEK : ^ 
D.VAN     NOSTRAND. 

1858. 


OTRODUCTION. 


The  first  part  of  tins  paper,  ending  with  Section  V.,  is  devoted  mainly 
to  the  theory  of  the  arch  first  proposed  by  Coulomb,  and  subseqnently 
developed  by  Audoy,  Petit,  Poncelet,  and  other  French  authors.  New 
developments  and  illustrations  are  here  given,  and  new  and  extensive 
tables  have  been  added. 

The  thrust  may  be  due  to  the  tendency  of  the  upper  vousson-s  to  slide 
down  their  beds,  or  to  the  tendency  of  some  upper  segment  of  the  arch  to 
rotate  on  the  interior  edge  of  its  lowest  joint.  Both  of  these  tendencies 
are  investigated  as  questions  of  maxima ;  and  the  greatest  of  the  two 
resulting  forces  is  the  true  thrust. 

In  almost  all  practical  cases  this  true  thrust  is  due  to  rotation,  and  is 
developed  at  the  instant  of  equilibrium  preceding  rupture  and  fall.  For 
convenience  we  have  called  this  the  ultimate  thrust.  It  is  obviously  much 
less  than  the  actual  thrust  of  the  well-established  arch  ;  nor  has  any  relation 
between  the  two  been  hitherto  pointed  out. 

The  defective  character  of  this  theory  was  indicated  by  Coulomb,  and 
has  been  clearly  seen  by  many  authors.  Moseley,  Mery,  and  others,  have 
written  able  treatises  in  search  or  support  of  a  better  principle,  but  no  one 
seems  to  have  seized  the  final  idea,  or  combination  of  ideas,  which  places 
the  theory  of  the  actual  thrust  upon  a  perfectly  definite  mechanical  basis. 
We  have  before  us  an  "  Examen  historique  et  critique  des  principales  the- 
ories concernant  I'Equilibre  des  Voutes,"  by  Poncelet,  drawn  up  with  the 
characteristic  ability  and  learning  of  that  great  man,  in  which  no  allusion 
is  made  to  any  attempt  to  furnish  a  definite  and  exact  theory  of  the  actual 
thrust  of  circular  and  elliptical  arches.     Approximations   to  this  thrust, 

712630 


;[gS  INTRODUCTION. 

reifardiiiir  tlie  curve  of  pressure  or  curve  of  equilibrium  as  a  sort  of  cat- 
enarv,  and  the  intrados  as  nearly  parallel  thereto,  have  indeed  been 
attained  by  English  mathematicians  of  the  last  century,  and,  of  late,  far 
more  completely,  it  is  said,  by  Yvon  Yillarceau  ;  and  Carvallo,  in  a  beauti- 
ful and  hltrhly  practical  treatise,  published  in  the  Annales  des  Fonts  et 
C'haussees,  1853,  has  <,dven,  approximately,  the  actual  thrust  and  all  other 
elements  of  semicircular  and  elliptical  arches  surcharged  horizontally.  We 
sav  approximately,  for  Carvallo,  to  facilitate  calculation,  regards  the  joints 
of  the  arch  as  vertical,  instead  of  perpendicular  to  the  intrados,  and  determ- 
ines the  angle  of  greatest  thrust  by  supposed  rotations  of  the  upper  seg- 
ments upon  the  intrados,  instead  of  the  lowest  of-  the  two  curves  which 
divide  the  joints  into  three  equal  parts. 

The  last  part  of  this  paper,  beginning  with  Section  VI.,  furnishes  an 
oriLjinal  and  probably  a  new  theory  of  the  arch,  with  numerous  tables  giv- 
ing the  actual  thrust  of  most  arches  in  common  use  without  calculation. 
This  actual  thrust  is  found  to  differ  largely,  in  most  cases,  from  the  ultimate 
thrust  above  mentioned,  their  ratio  varying  from  1  to  nearly  2. 

Tliis  new  theory  is  based  on  the  principle  that  the  curve  of  pressure 
shall  not  approach  the  intrados  or  extrados  within  certain  prescribed  limits, 
and  that  it  shall  touch,  at  the  three  or  five  joints  of  rupture,  so-called,  the 
two  curves  which  pass  through  those  limits.  Every  case,  as  in  the  ultimate 
thrust,  is  a  question  of  maxima. 

Coulondt  pointed  out  the  several  modes  of  rupture  to  which  arches  are 
liable  ;  in  one  of  which  the  crown  rises.  Most  writers  upon  the  arch  repeat 
his  remarks;  but  no  one  seems  to  have  given  the  subject  any  particular 
consideration — much  less  to  have  determined  the  limits  within  which  such 
rupture  is  possible  or  probable.  Indeed,  without  a  knowledge  of  the  actual 
thrust  no  practical  or  useful  solution  was  possible.  The  reader  will  be 
surprised  to  learn  that  most  of  the  light  beautiful  stone  bridges  of  Great 
I^itain  are  inclined  to  this  mode  of  rupture.  Our  remarks  on  this  subject, 
in  relation  to  the  limits  of  possible  and  practicable  arches,  and  in  the  discus- 
eion  of  Table  I,  w  ill  be  found,  we  trust,  both  interesting  and  useful.  Eveiy 
proposed  bridge  of  large  span  should  be  investigated  in  view  of  this  third 
mode  of  rujtture,  in  which  the  arch  may  fall  without  disturbing  the  piers. 
If  not  unnecessarily  heavy,  the  bridge  will,  in  general,  be  in  the  neighbor- 
hoo<l  of  the  prncticahility  connected  with  this  mode  of  rupture. 

We  have  taken  great   pains   to   prepare  simple  and  comprehensive 


INTRODUCTION.  189 

formulre  for  thickness  of  piers.  These  formukx)  are  the  same  in  form, 
whether  we  use  the  actual  or  ultimate  thrust ;  the  only  difference  being  in 
the  magnitude  of  the  thrust,  and  in  its  point  of  application  or  lever  arm. 

Both  the  ultimate  and  actual  thrusts  of  many  arches  consist  of  two 
parts  :  first,  the  thrust  of  the  arch  proper  and  a  certain  part  of  its  load ; 
second,  the  effect  of  a  surcharge  of  constant  vertical  depth. 

These  effects  have  been  investigated  separately,  and  the  two  maxima, 
taken  directly  from  the  tables,  give,  Avhen  added  together,  the  entire  thrust 
in  very  slight  excess.  The  accuracy  of  this  method  may  be  proved  by 
comparing  the  results  of  Table  F  with  those  of  a  table  given  by  Moseley  at 
the  end  of  his  "  Mechanics  of  Engineering." 

Although  the  tables  and  the  reduced  formulae  provided  for  the  actual 
thrust  cover  most  of  the  cases  likely  to  occur  in  practice,  still  the  subject 
admits  of  further  development  and  illustration,  as  the  field  is  supposed  to 
be  entirely  new.  Additional  tables  are  desirable  to  make  the  theory  more 
completely  available,  and  some  of  the  given  tables  require  enlargement. 

In  connection  with  this  theory  we  have  said  nothing  of  the  sliding 
thrust ;  because  that  had  already  been  disposed  of  in  connection  with  the 
ultimate  thrust. 

The  sliding  thrust,  when  it  exceeds  the  actual  rotation  thrust,  is,  how- 
ever, given  in  the  tables  at  the  end  of  Section  VI.  Cases  of  this  kind  are 
very  rare. 

Very  few  arches  conform  precisely  to  the  conditions  which  we  are 
compelled  to  assume  in  the  preparation  of  tables  and  formulae.  The  geo- 
metrical method,  given  at  the  end  of  the  paper,  is  independent  of  all  con- 
ditions, and  will  furnish  in  a  short  time  all  the  elements  of  the  most  com- 
plex case.  It  affords  facilities  for  discussing  the  plan  of  an  arch,  which  can 
hardly  be  found  in  calculation  alone. 

It  can  be  modified  so  as  to  give,  by  successive  approximations,  an 
intrados  and  an  extrados  at  equal  distances  from  the  curve  of  pressure, 
Avith  joints  proportional  to  the  pressure  upon  them  ;  the  pressure  per  unit 
of  surface  on  these  joints  being  constant  throughout  the  arch,  and,  if  we 
desire  it,  throughout  the  pier,  which  may  always  be  treated  as  a  part  of 
the  arch.  Tlie  segmental  arch,  bounded  by  arcs  of  different  circles,  is 
itself,  in  most  cases,  a  good  solution  of  this  problem  of  least  material. 

The  curve  of  pressure,  in  the  same  arch,  at  different  stages  of  its  load, 
is  liable  to  great  changes.     Provided  the  extreme  variations  be  allowable, 


IQQ  INTRODUCTION. 

compariiiij  the  unloa<lc(l  with  the  fully  loaded  arch,  all  iiitLM-mediate  varia- 
tions mav  be  kept  still  nearer  the  mean  by  putting  on  the  load,  in  due  pro- 
portion, at  the  several  parts  of  the  arch,  simultaneously.  Wo  are  indebted 
for  this  idea  to  M.  Carvallo  who  niakos  perhaps  extreme  use  of  it. 

To  prevent  any  change  at  all  in  the  curve  of  pressure,  in  the  arch  with- 
out anv  load  and  the  same  arch  fully  loaded,  he  goes  so  for  as  to  make  the 
depth  of  the  arch  in  constant  proportion  to  the  depth  of  the  load  on  the 
same  vertical  lines.  This  may  be  very  proper  in  aqueduct  and  other  heavy 
bridrt-es ;  but  in  li<>-ht  arches  without  much  load  at  the  key — the  case  of 
common  bridges — it  would  give  an  oxtrados  nearly  horizontal.  Certainly, 
there  is  no  objection  to  a  change  in  the  situation  of  the  curve  of  pressure, 
provided  it  remain  within  the  limits  prescribed  by  Xavier ;  since,  in  that 
case,  the  change  cannot  be  for  the  worse,  and  by  allowing  it  we  greatly 
diminish  the  quantity  of  material,  without  in  any  degree  impairing  the 
stability  of  the  work. 

No  attempt  has  been  made  to  investigate  the  change  of  form  and  cir- 
cumstances which,  in  some  measure,  must  always  result  from  the  compres- 
sion of  the  material.  If  we  suppose  the  piers  immovable,  the  extent  of 
joint  everywhere  proportional  to  the  pressure,  and  the  curve  of  pressure 
everywhere  at  the  middle  of  the  joint,  compression  w  ill  conveyt  the  circu- 
lar into  an  cUi[>tical  arch — a  change  in  most  cases  favorable  to  the  stabil- 
ity of  the  light  arch,  although  it  increases  the  thrust.  These  conditions 
can  only  be  fully  realized  under  a  very  peculiar  load  ;  they  will,  however, 
often  be  nearly  realized,  and  will  cause,  approximately,  the  cliange  above 
mentioned. 

In  connection  with  the  actual  thrust.  Section  VI.,  nothing  is  said  of 
adhesion  of  mortar,  because  this  force  can  never  act  in  the  well  propor- 
tioned arch,  the  curve  of  pressure  passing  within  such  limits  that  every 
part  of  every  joint  is  compressed. 

The  reader  who  imiy  consult  this  paper  for  practical  pui*poses  only,  is 
advised  to  begin  with  Section  VI.,  and  to  refer  to  the  first  part  of  the  work 
only  when  necessary  to  understand  the  second.  He  is  also  advised,  as  the 
arch  must  be  regarded  bom  inaiiy  points  of  view,  to  accpiire  a  general 
knowledge  of  the  whole  of  that  Section,  before  studying  critically  the 
particular  parts. 

It  may  be  a'^ked,  AVliy  not,  then,  invert  the  work,  and  place  the  more 
important  part  fii-st  ?      The  reason  will   be   given.      I'hc  first  part   was 


INTRODUCTION.  "  191 

nearly  completed  before  the  author  had  discovered  an  exact  mechanical 
foundation  for  a  theory  of  the  actual  thrust— a  foundation  on  which  tables 
of  tliat  thrust  could  be  calculated,  and  definite  results  obtained,  Avithout 
any  sacrifice  of  principle  to  facility  of  calculation.  The  second  part,  con- 
trary to  the  original  intention,  began  to  expand,  and  finally  assumed,  well 
nigh,  a  character  of  completeness  by  itself.  It  was  too  late  to  re-write  the 
vhole.  The  author  had  neither  time  nor  health  to  do  so.  Noi-  could  the 
fitst  part  be  dispensed  with  :  it  is  necessary  as  a  complement  to  the  sec- 
ond It  is  to  be  regretted,  however,  that  the  fullness  of  illustration  and 
demonstration  given  to  the  first  part  of  the  work  has  not  been  given  rather 
to  the  second. 

The  great  variety  of  dimensions  adopted  by  different  engineers  for 
arches  of  nearly  equal  span  and  rise,  is  shown  by  Table  I.  No  better 
proof  could  be  given  of  the  utility  of  some  little  attention  to  the  theory  of 
the  arch.  The  engineer  may  thereby  avoid,  on  the  one  hand,  a  large 
wasteful  expenditure  for  useless  excess  of  strength,  or,  at  the  otlier  extreme, 
the  mortification  of  seeing  his  work  fall  down  in  consequence  of  impossible 
proportions. 

The  preparation  of  this  paper  has  involved  long  and  arduous  labor,  in 
the  reduction  of  formulaa  to  numerical  forms,  and  in  the  calculation  of 
tables.  The  author  gratefully  acknowledges  the  assistance  of  Dr.  B.  "W. 
Whitehurst  and  Mr.  George  B.  Phillips,  in  making  the  numerical  cal- 
culations. Without  their  efficient  aid,  he  could  not  have  undertaken  so 
larp-e  a  taslc. 


CONTENTS. 


[Such  parts  as  are  supposed  to  be  entirely  new,  are  given  in  italics.      The 
numbers  refer  to  the  paragraphs.] 


SECTION  I. 


GENERAL    INVESTIGATION    WITHOUT    REGARD    TO    PARTICULAR    CURVES. 

Definitions  and  general  remarks,  1 ;  influence  of  common  and  hydraulic 
mortar,  2 ;  general  discussion  of  the  thrust  in  the  first  mode  of  rupture,  by- 
rotation,  .3—11  ;  stability  of  the  pier,  11,12;  eftect  of  surcharge  upon  the 
ultimate  thrust,  13,  14  ;  the  effect  upon  the  thrust  of  any  supposed  ad- 
hesion of  mortar  at  the  joint  of  rupture  and  at  the  vertical  joint,  15  ; 
general  formuliB  for  the  ultimate  thrust  in  the  first  and  common  mode  of 
rupture,  by  rotation,  and  for  thickness  of  pier,  17;  the  coefficient  of 
stability,  as  used  by  Audoy,  Poncelet,  and  others,  18,  19;  general  discus- 
sion of  the  thrust  due  to  sliding,  20—22  ;  point  of  application  of  the  thrust, 
23  ;  the  manner  of  expressing  the  eftect  of  adhesion,  24  ;  general  discussion 
of  the  third  mode  of  rupture,  by  rotation,  the  key  rising,  25  ;  fourth  mode 
of  rupture,  sliding,  2G.     Page  199—215. 


194  CONTENTS. 


SECTION   II. 


THE    SEMICIRCULAR    ARCH    OF    EQUAL    THICKNESS    THROUGHOUT. 

Application  of  the  general  formulic,  27 — 47;  arches  without  rotation 
thrust,  30;  effect  of  mortar,  31,  32;  comparative  effect  of  mortar  upon 
lurfje  and  small  arches,  33  ;  effect  of  surcharge  of  a  constant  depth,  34  ; 
the  entire  ultimate  thrust  obtained  by  addinr/  two  nuixima  together,  34 ;  the 
effect  of  a  column  or  weight  upon  the  key  of  the  arch,  35  ;  the  thrust  due  to 
sliding,  3G — 39 ;  the  entire  sliding  thrust  obtained  by  adding  two  maxima 
together,  39;  the  true  thrust,  41 ;  thickness  of  pier,  forniuhe,  42  ;  limit 
thickness  of  piers,  43 ;  discussion  of  table  A,  46  ;  limit  thickness  of  the 
circular  ring,  47,     Page  215 — 234. 


SECTION   III. 


THE  MAGAZINE  ARCH,  INCLUDING  THE  EXTREME  CASE  IN  WHICH  THE  TWO 
PLANES  WHICH  FORM  THE  ROOF,  ARE  ONE  AND  HORIZONTAL,  OR  THE 
CIRCULAR    ARCH    SURCHARGED    HORIZONTALLY. 

Application  of  the  general  formulic  to  the  magazine  arch,  48 — 67  ;  the 
ultimate  thrust,  due  to  rotation,  48 — 51  ;  effect  of  surcharge  of  a  constant 
depth,  50 ;  the  entire  rotation  thrust  obtained  by  adding  two  maxima 
together,  50  ;  effect  of  a  single  column  resting  upon  the  ridge  of  the  arch,  51 ; 
the  .sliding  thrust,  52 — 54  ;  the  entire  sliding  thrust  obtained  by  adding  two 
maxima  togetJier,  54 ;  remarks  on  tables  C,  D,  F,  54,  55  ;  the  roof  inclined 
45°,  50;  the  roof  horizontal,  57  ;  discussion  of  table  C,  the  roof  inclined 
45°,  58;  table  D,  the  roof  horizontal,  59;  discussion  and  use  of  table  F, 
which  gives  directly,  or  by  proportional  parts,  the  ultimate  rotation  thrust 
ff  all  circular  magazine  arches  in  common  use,  with  and  tvithout  surcharge, 
aim  the  sliding  thrust  xchen  this  exceeds  the  other,  GO — 63  ;  thickness  of 
piers,  universal  methoil,  64  ;  jmrticular  cases,  65,  66  ;  examples,  67  ;  limit 
thickness  of  piers,  64.     Vivfa  235 — 264. 


CONTENTS.  105 


SECTION   IV. 


SEGMENTAL    ARCHES — SCARP    WALLS. 


General  remarks,  G8 ;  the  segmental  ring,  69 — 73  ;  thickness  of  pier, 
72,  73  ;  example,  the  Monocacy  bridge,  73  :  segmental  arches  sm-charged 
horizontally,  74 — 79  ;  thickness  of  pier  ^  79  ;  segmental  arches  with  inclined 
roofs,  80 — 82  ;  thickness  of  jjier,  82  ;  stability  of  a  scarp  ivall,  83,  84  ;  seg- 
mental arches,  ap)p)roximate  formulce,  85 — 89.     Page  264 — 290. 

SECTION   V. 

ELLIPTICAL    ARCHES. 

General  remarks,  90 ;  the  unloaded  eUip)tical  ring^  91 — 95  ;  the  ellipti- 
cal arch  ivith  elevated  ridge^  96 — 98  ;  elliptical  arches  surcharged  horizon- 
tally^ 99 — 107  ;  at  the  end  of  section  V.  tables  A,  B,  C,  D,  E,  from  volume 
12  Memorial  de  I'officier  du  Genie  ;  and  tables  E',  F,  G,  II,  prepared  for 
this  paper ;  these  tables  give  the  ultimate  rotation  thrusts  or  the  sliding 
thrusts  when  the  latter  exceed  the  former.     Pao-e  290 — 323. 


SECTION  VI. 

THE    CURVE  OF    PRESSURE,  NEW  THEORY  OF    THE    ACTUAL  THRUST,  ETC.  ETC., 
BASED    ON    KNOWN    MECHANICAL    PRINCIPLES. 

[The  whole  of  this,  as  a  definite  system,  is  supposed  to  be  new :  the  particulars 
are  not  given  in  italics.] 

Objections  to  the  theory  of  the  ultimate  thrust,  108 ;  general  prin- 
ciples, 108 — 111 ;  pressure,  per  unit  of  surface,  on  the  joints,  112  ;  funda- 
mental principles  of  the  theory  of  the  actual  thrust,  111 — 113;  actual 
thrust  of  the  unloaded  circular  ring,  114;  thrust  of  semicircular  arches 
surcharged  horizontally,  115,  116,  118;  the  magazine  arch,  117;  segmen- 
tal arches  surcharged  horizontally,  119,  120;  elliptical  arches  surcharged 
horizontally,  121 ;  the  coefficient  of  stability,  122 — 126  ;  thickness  of  pier, 
universal  method,  128  ;  particular  cases,  129 — 134  ;  examples,  135  ;  thick- 
ness of  arch,  how  determined,  example,  the  Monocacy  bridge,  136  ;  proper 
increase  of  thickness  at  and  below  the  joint  of  greatest  thrust,  137 — 139  ; 
relative  pressure,  per  unit  of  surface,  at  the  key  and  at  the  weakest  joint 
below  the  key,  140  ;  pressure,  per  unit  of  surface  on  the  joints  of  the  pier, 
141  ;  third  mode  of  rupture,  by  rotation,  the  key  rising,  142  ;  limit  thick- 
ness  of    possible    circular    arches   surcharged    horizontally,    142  ;    limit 


190  CONTENTS. 

thickness  of  possible  segmental  arclies  surcharged  horizontally,  143;  limit 
thickness  of  practicable  circular  arches  surcharged  horizontally,  144  ;  ditto, 
segmental  arches  surcharged  horizontally,  145;  cftect  of  surcharge  upon 
the  practicability  of  arches,  140  ;  equation  of  the  curve  of  pressure  in  the 
arch,  147  ;  point  of  application  of  the  actual  thrust  at  the  key,  148  ;  point 
of  application  of  the  ultimate  thrust  at  the  key,  149;  remarks  on  table  I. 
150  ;  "eneral  remarks  upon  the  determination  of  the  thrust  and  upon  the 
tluckness  of  the  arch  at  the  key  and  at  the  weakest  joints,  151,  152;  geo- 
metrical methods  of  universal  application,  153 — 159;  rupture  of  masonry 
by  compression,  160  ;  curve  of  pressure,  in  the  pier,  IGl  ;  in  the  arch,  162. 
raire  324—423. 


TABLES 

OF    THE    ULTIMATE    THRUST,    ETC. 

A.  Semicircular  arches,  unloaded  :  ultimate  rotation  and  sliding  thrusts 
limit  thickness  of  piers,  etc.  etc.,  page  310-11  ;  explanation,  paragraph 
40. 

13.  Formulte  for  thickness  of  pier,  unloaded  semicircular  arches,  page 
312-13;  explanation,  paragraph  42. 

C.  Magazine  arches,  the  roof  inclined  45°  :  ultimate  rotation  and  sliding 

thrusts,  etc.,  etc.,  page  314-15  ;  explanation,  par.  58. 

D.  Semicircular  arches  surcharged  horizontally:  ultimate  rotation  and  slid- 

ing thrusts,  page  316-17  ;  explanation,  par.  59. 

E.  Tlie  unloaded  segmental  ring ;  seven  systems  :  the  ultimate  rotation,  or 

the  sliding  thrust;  in  all  cases  the  greatest  of  the  two,  page  318  ;  ex- 
planation, par.  09. 

E'.  Segmental  arches  surcharged  horizontally;  seven  systems:  the  ultimate 
rotation,  or,  the  sliding  thrust;  in  all  cases  the  larger  of  the  two,  page 
319;  explanation,  par.  74. 

F.  Magazine  arches,  ten  systems,  with  columns  for  surcharge:  the  ultimate 

rotation,  or,  the  sliding  thrust;  in  all  cases  the  larger  of  the  two,  page 
320-1,;  explanation,  par.  00 — 63. 

(1.  Elliptical  arclies  surcharged  horizontally  :  the  ultimate  rotation  thrust  in 
eight  systems;  tlie  sliding  thrust,  being  less  than  the  former,  is  not  in 
any  ciisc  given, — page  322  ;  explanation,  par.  99. 


CONTENTS,  lOV 

H.  Elliptical  arches  witliout  load  :  the  ultiinate  rotation  tlirnst  in  t^vo  sys- 
tems ;  the  sliding  thrust,  being  less  than  the  former,  is  not  given, — 
page  323  ;  explanation,  par.  93. 

A'.  Corrections,  to  be  used  in  computing  the  ultimate  thrust  of  the  un- 
loaded elliptical  arch  from  table  A,  page  294,  par.  92. 

Table  of  the  ultimate  thrusts  of  elliptical  and  segmental  arches,  compared, 
page  304  ;  par.  101. 

Table  of  additions  to  the  ultimate  rotation  thrust  of  elliptical  arches  of  all 
kinds,  caused  by  a  surcharge  of  constant  depth,  page  306  ;  par.  104. 


TABLES 


OF    THE    ACTUAL    THRUST,    ETC.    ETC. 

AA.  Semicircular  arches,  unloaded :  ultimate  and  actual  thrusts,  ratio  of 
the  two ;  the  sliding,  being  less  than  the  actual  rotation  thrust,  is  not 
given, — page  424;  explanation,  par.  114. 

DD.  Semicircular  arches  surcharged  horizontally,  the  curve  of  pressure 
starting  at  the  middle  of  the  key  and  passing  through  the  middle  of 
the  joint  of  greatest  thrust :  the  ultimate  and  actual  rotation  thrusts 
and  the  ratio  of  the  two;  the  ultimate  and  actual  eifects  of  surcharge 
also  compared ;  the  sliding,  being  less  than  the  actual  rotation  thrust, 
is  not  given, — page  425  ;  explanation,  par.  116. 

FF.  Magazine  arches,  four  systems  :  actual  rotation  thrusts,  with  a  column 
for  surcharge  ;  coefficient  of  stability  or  ratio  of  the  actual  to  the  ulti- 
mate thrust;  the  sliding,  being  less  than  the  actual  rotation  thrust,  is 
not  given, — page  426-7  ;  explanation,  par.  117. 

DDD.  Semicircular  arches  surcharged  horizontally,  the  curve  of  pressure 
starting  at  one  thii'd  the  length  of  the  joint  from  the  extrados  at  the 
key,  and  passing  the  weakest  joint  at  the  same  distance  from  the  in- 
trados  :  actual  rotation  thrusts  with  a  column  for  surcharge ;  angles 
of  greatest  thrust,  on  three  different  hypotheses,  compared  ;  coeffi- 
cients of  stability,  compared;-  the  sliding,  being  less  than  the  actual 
rotation  thrust,  is  not  given, — page  428  ;   explanation,  par,  118. 


lOS  COXTEN're. 

EE.  Segmental  artlie>  surcharged  horizontally :  actual  thrust  in  seven  sys- 
tems, with  columns  giving  the  effect  of  surcharge,  and  the  ratio  of  the 
actual  to  the  ultimate  thrust,  or  the  coefficient  of  stability  ;  the  sliding, 
there  exceeding  the  actual  rotation  thrust,  is  given  in  the  upper  part 
of  the  last  column, — page  430-1  ;  explanation,  par.  120. 

I.  Elements  of  celebrated  bridges,  page  432-3  ;  explanation,  par.  150. 

Table.  Limit  thickness  of  possible  segmental  arches,  page  385  ;  par.  143. 

Table.  Limit  thickness  of  practicable  segmental  arches^page  389  ;  par.  145. 


E 11  R  A  T  A  . 


(1  (f 

Page-  a?:,  line  14  fcoiii  top,  for    A'=—    n;i(l    K=\-\ — . 


1  5 

Page  310,  over  2d  coluuiii,  fur    r—hf  road   r—^f. 


THE 


THEORY  OF  THE  ARCH. 


SECTION  I. 


DEFI]SriTIO:N"S    AND    GENERAL    RE^fARKS. 

1.  The  arcli  proper  consists  of  several  parts,  commonly 
called  voussoirs^  wliicli  press  upon  eacli  other  in  surfaces 
meeting  at  one  or  more  central  lines.  It  is  distinguished 
from  the  beam  by  exerting,  upon  its  piers  or  abutments,  an 
outward  thrust. 

It  may  be  upright  or  reversed,  horizontal  or  inclined. 

Cylindrical  arches  are  fully  represented  by  a  section 
taken  at  right  angles  to  their  general  direction.  The  sur- 
faces of  this  section  give  the  relative  masses  of  all  the  parts 
of  the  arch  and  its  piers ;  and  to  obtain  the  actual  weight 
of  a  unit  in  width,  it  is  only  necessary  to  multiply  the  sur- 
face, expressed  in  feet  or  any  other  unit,  by  the  weight  of 
a  cubic  unit  of  masonry. 

The  arch,  when  sul:)mitted  to  calculation  for  the  pur- 
pose of  determining  the  recpiisite  dimensions  of  its  piers 
and  other  parts,  is  supposed  to  be  a  unit  in  width. 

The  extradO'S  is  the  exterior  outline  of  the  arch  proper, 
whether  made  uj)  of  curves  or  straight  lines. 
2 


OQQ  THEORY    OF    THE    ARCH. 

The  intrados  is  tlie  interior  line,  and  tlie  corresponding 
?urfi\ce  of  the  arch  is  tlie  soffit. 

The  wedge-like  pieces  of  which  the  arch  is  composed, 
are  called  voussoirs. 

Ill  this  paper  it  will  be  taken  for  granted,  unless  other- 
wise specially  mentioned,  that  all  arches  have  a  joint  at 
the  summit,  midway  over  the  oi)eniiig  between  the  piers, 
called  the  key-stone  or  vertical  joint. 

The  sides  of  the  arch  are  called  the  reins. 

The  line  or  bed  at  whicli  the  arch  begins,  or  springs 
from  its  piers,  is  called  the  springing  line. 

The  Itlocks  of  masonry,  or  other  material,  which  sup- 
port two  successive  arches,  are  called  piers  ;  the  extreme 
blocks,  whicb  generally  support,  on  one  side,  embank- 
ments of  earth,  are  called  ahutments. 

A  pier  strong  enough  to  withstand  the  thrust  of  either 
adjacent  arch,  should  the  other  fall  down,  is  sometimes 
called  an  ahutment pier. 

Besides  their  own  weight,  arches  usually  support  a 
permanent  load  or  surcharge  of  earth  or  masonry,  or  both. 

2.  Common  mortar  in  heavy  masses  of  masonry,  har- 
dens very  slowly — remaining  comparatively  soft  long  after 
the  centerinsr  has  been  removed  and  the  arch  left  to  its 
own  sui)port.  "VVe  must  not,  therefore,  in  our  calculations 
for  large  arches,  look  for  any  element  of  strength  or  sta- 
bility in  the  adhesion  of  mortar  made  of  common  lime. 
We  must  suppose  the  voussoirs,  whether  large  or  small,  to 
press  \\\M)\\  eacli  other  witliout  adhesion  to  resist  either 
rotation  <.)r  sliding.  Tlie  joints  are  always  rough,  and  any 
tendency  to  sliding  is  resisted  by  ordinary  friction. 

On  the  other  hand,  arches  of  all  sizes  laid  hi  good 
liy<b-auHc  mortar,  and  tliiii  arches  laid  in  common  mortar, 
may  derive  some  incretuse  of  stability  from  the  adhesion  of 
the  mortar  which  unites  the  joints. 

Fortunately  we  can,  by  calculation  or  geometrical  con- 


ROTATION.  201 

stniction,  easily  determine  tlie  whole  effect  of  tliis  adhe- 
sion, when  we  assign  to  it  a  particular  value,  that  is,  so 
many  pounds  per  square  inch. 

Perfectly  flat  arches,  with  vertical" joints,  are  entirely 
supported  by  this  adhesion.  They  are  not  properly  arches, 
but  beams. 

So  far  as  any  arch  derives  strength  from  the  adhesion 
of  its  parts  to  each  other,  it  partakes  of  the  nature  of  a 
beam. 

The  thrust  of  the  arch,  due  to  rotation  in  the  case  of 
actual  rupture  and  fall,  is,  for  convenience,  sometimes 
called  the  ultimate  thrust. 

THEORY    OF    COULOMB. 

3,  We  shall  now  indicate,  in  its  most  general  terms,  the 
theory  of  the  arch  first  proposed  by  Coulomb,  and  subse- 
quently developed  l)y  Audoy,  Petit,  Poncelet,  and  other 
distinguished  French  engineers  ;  a  theory  hitherto  emin- 
ently useful,  notwithstanding  its  great  defects. 

We  shall  hereafter,  under  the  head  of  curve  of  pres- 
sure^ notice  these  defects,  and  try  to  give  a  more  exact  and 
practical  theory  of  the  circular,  segmental,  and  elliptical 
arch. 

FIRST    AND    ORDINARY   MODE    OF   EUPTURE ROTATION. 

4.  Let  figure  1,  plate  10,  represent  a  section  of  any 
symmetrical  arch  in  a  condition  of  stability.  We  may 
suppose  this  to  be  composed  of  two  equal  half-arches, 
meeting,  and  mutually  supporting  each  other,  at  the  verti- 
cal or  key-stone  joint.  From  the  perfect  equality  of  the 
two  halves,  it  is  evident  that  the  pressure  of  the  one  is 
counterl)alanced  by  the  resistance  of  the  other,  and  that 
the  two  forces  are  equal,  horizontal,  and  directly  opposed. 

This  mutual  pressure  is  exerted,  generally,  all  along  the 
vertical  joint,  f]*om  the  intrados  to  the  extra dos.     Its  re- 


202  THEORY     OF    TDE    ARCH. 

snltant  i>;  api)lie(l  at  -jome  iinknowu  point  ordinarily  alxjve 
tlu*  middle  of  that  joint.  The  position  of  the  resultant, 
howt'ver,  becomes  known,  as  we  shall  soon  see,  when  the 
anil,  no  lonirer  al>le  to  maintain  itself,  begins  to  fall,  from 
insufficient  resistance  in  itself  or  its  supports. 

I^'t  fiir.  -  represent  a  section  of  the  arch  in  a  condition 
of  in^tahility — that  is,  all  Init  aide  to  stand  upon  its 
springing  lines,  and  just  Ijeginning  to  fall.  Fig.  3  repre- 
sents the  piers  as  adhering  to  the  lower  parts  of  the 
arch,  and  the  whole  as  just  l)eginning  to  fall.  This  is 
hy  far  the  most  common  mode  of  ru}>ture, — almost  the 
only  mode,  according  to  the  French  authors  cited  above, 
to  ^^'hieh  the  cii'cular  arch  of  common  use  is  at  all  exposed. 

5.  In  this  first  mode  of  rui)ture,  the  arch  divides  into  four 
pieces;  the  crown  settles  down;  the  reins  spread  out;  the 
vertical  joint  opens  at  the  intrados,  the  adjacent  segments 
touching  only  at  the  extrados ;  the  reins  open  at  the  extra- 
dos,  the  adjacent  segments  touching  only  at  the  intrados  ; 
the  weakest  joint  below  the  reins  opens  on  the  ijiside. 

The  two  lower  segments  revolve,  outwardly,  on  the 
exteri«jr  edge  of  Avhat  Ave  have  called  the  weakest  joint 
below  the  reins — thereby  leaving  room  for  the  two  u])i»er 
segments  to  revolve,  towards  each  other,  on  the  interi(jr 
edges  of  the  joints  at  the  reins.  If  the  first  rotation  be 
jn-evented,  l>v  the  weight  of  the  piei's  or  lower  parts  of 
the  ai*-li,  the  second  rotation  will  also  be  prevented ;  for 
the  crown  can  not  settle  down  unlf'ss  the  reins  spread 
out. 

Wliat  we  have  called  the  weakest  joint  below  the 
reins,  that  is,  the  joint  most  disposed  to  open  under  the 
hoiizontal  foix-e  or  pre.^suie  acting  at  the  crown,  is,  in 
almost  all  ju'actical  cases,  the  joint  at  the  base  of  the 
]»iei',  or  at  the  springing  line  if  there  be  no  pier. 

('..   Let   US  supj)ose  one  half  of  the  arch  removed,  fig. 
\.  ;ind  a]ij)ly  at  (f,  the  highest  point  of  the  arch  proper,  a 


ROTATION.  203 

lioi'izontal  force,  F^  just  sufficient  to  keep  tlie  i-eiiiaining 
semi-arcli  in  its  place.  This  force  is  tlie  tlivu-st  of  the  ai-ch, 
and  is  equivalent  to  the  mutual  pi-essure  which  before 
existed.  It  is  evident  that  the  arch,  in  its  tendency  to  fall, 
cannot  generate  a  pressure  or  force  greater  than  that 
which  will  prevent  its  falling. 

7.  If  the  semi-arch  were  in  one  solid,  indivisible  piece, 
standing  on  its  springing  line,  A  B^  the  force  F  Avould 
be  known  at  once  ;  for,  calling  Jf  the  moment  of  the  semi- 
arch,  in  reference  to  J,  the  interior  edge  of  the  springing 
line,  we  should  have 

M 

Fx  a  C—  21;  or  i^=  —,y. 
'  at 

8.  But  the  arch  is  not  solid :  it  may  separate  at  any 
joint.  If  we  regard  any  segment  of  the  arch  a  h  m  n  r  a^ 
fig.  4,  we  see  that  a  certain  force,  F\  is  necessary  to  pre- 
vent its  fall,  or  rotation  forwards,  on  the  inner  edge  of  the 
joint  m  n. 

From  <7,  the  center  of  gravity  of  the  segment  of  the 
arch  proper  and  its  surcharge,  droji  the  vertical  line  g  g 
upon  the  horizontal,  m  m'. 

Let  the  surface  a  h  m  n  r  a:=S. 
"  lever  arm  a  wz'=y. 

We  then  have  the  equation  of  moments 
F'y=Si,;  or  r='^; 

9.  ^ow,  if  we  suppose  the  position  of  the  joint  m  n  to 
vary,  in  succession,  from  the  key  to  the  springing  line,  the 
force  F\  required  to  prevent  the  fall  of  the  corresponding 
segments,  will  also  vary.  It  will  be  very  small  near  the 
summit,  gradually  increase  towards  the  reins,  become  a 
maximum  about  50  or  60  degrees  from  the  key,  and  grad- 


204  THEORY    OF    THE    ARCH. 

iinllv  (limiiiisli  to  tlie  springing  line.  Tliis  maximum  value 
of  F\  whic'li  we  shall  call  F,  is  the  thrust  of  the  arch. 
As  it  is  sufficient  to  sustain  that  segment  which  required 
the  greatest  support,  it  is  more  than  sufficient  to  sustain  all 
other  segments,  including  the  semi-arch  itself. 

10.  The  magnitude  of  this  thrust,  and  the  position  of 
the  joint  corresponding  to  the  angle  of  maximum  thrust, 
are,  it  is  ea.<y  to  see,  entirely  unaffected  by  the  piei-s  and 
the  loAver  })art8  of  the  arch, — that  is,  by  anything  below 
the  joint  itself. 

This  remark  is  important,  for  it  announces  one  of  the 
few  simple  i)rinciples  on  which  the  whole  theory  of  the 
arch  depends. 

11.  The  tendency  of  the  horizontal  thrust  is  to  push 
over  the  semi-arch,  causing  it  to  turn  round  the  exterior 
edge  of  its  lowest  joint.  But  if  the  moment  of  the  thrust 
taken  in  reference  to  that  center  of  rotation,  be  less  than  thei 
moment  of  the  semi-arch  and  pier,  in  reference  to  the  same 
point,  no  motion  can  ensue :  the  arch  is  stable.  On  the 
other  hand,  if  the  moment  of  the  thrust  prevail,  the  arch 
will  begin  to  fall  as  represented  in  figures  2,  3. 

The  equation  of  moments  will  be 

M  representing  the  sum  of  the  moments  of  the  semi-arch 
an<l  it>  jtier,  and  /  the  lever  arm  of  the  thrust. 

1"2.  If  we  take  into  consideration  the  adhesion  of  mor- 
tar up<»n  tlic  joint  of  the  springing  line,  or  base  of  the 
]>ier,  the  e([uati()n  <>f  moments  will  become,  art.  15  and 
followinir, 

f  rcpi'esentiiig  tin-  foi'ce  of  adhesion  upon  a  imit  of  surface, 
and  e  the  length  of  the  lower  joint.     In  this  expression 


ROTATION".  205 

there  is  notliing  unknown  except  tlie  tliiekness  of  the  pier, 
which  enters  into  M. 

13.  If  the  ai'ch  have  any  surcharge  whatever,  not 
already  inchided  in  S^  we  can  always  represent  its  weight 
by  a  surface  of  masonry.  Let  8'=aas8^  fig,  4^,  represent 
such  w^eight  in  magnitude  and  position;  and  let  79'  repre- 
sent the  horizontal  distance  of  its  center  of  gravity  from 
the  vertical  passing  through  m^  the  center  of  rotation. 
The  horizontal  force  necessary  to  prevent  the  rotation  of 
any  particular  segment  and  its  load,  will  now  be 

y     y  ' 

The  maximum  value  of  F'  as  before,  art.  9,  will  be  the 
true  thrust. 

We  might,  in  arches  of  large  span,  wish  to  ascertain 
the  effect,  upon  the  thrust,  of  several  weights  in  given 
positions.  Let  8\  S"  &c.,  represent  their  several  magni- 
tudes ;  2)\  2^"  <fec-i  ^^^  respective  distances  of  their  centers 
of  gravity  from  the  vertical  line  through  m.     We  have 

y 

14.  In  general,  however,  the  surcharge  is  continuous, 
and  bounded  l^y  one  or  more  straight  lines. 

It  is  hardly  necessary  to  say  that,  in  the  equation  above 
given,  the  weights  8\  8",  &c.,  are  supposed  to  be  located 
over  the  segment  a  h  m  n  r  a,  to  wdiich  F'  relates  ;  and 
that,  if  the  center  of  gravity  of  any  one  of  them,  as  8",  is 
on  the  left  of  the  vertical  through  m,  the  lever  arm,j?", 
will  be  negative,  and  the  product  8"p\  therefore, negative. 

The  surcharge  adds,  generally,  but  little  to  the  difficulty 
of  the  investigation,  as  will  hereafter  be  seen. 

15.  To  complete  the  general  formulae  for  the  first  and 
most  common  mode  of  rupture,  it  is  necessary  to  add 
expressions  for  the  adhesion  of  mortar  upon  the  joints. 


206  THEORY     OF    THE    ARCH. 

The  ultimate  resistance  of  mortar  to  compression  is  much 
o-reater  than  its  resistance  to  extension.  Still,  we  can, 
without  error,  re^irard  these  forces  as  equal,  provided  we 
determine  their  value  by  expei-iments  upon  rectangular 
prisms  of  the  same  material  as  the  arch, — that  is,  by  ob- 
serving the  weight  necessary  to  break  a  beam  supported 
at  one  or  both  ends.  The  product  of  this  Aveight  by  its 
lever  arm  makes  known  the  effective  resistance  opposed 
by  the  mortar.  The  expression  |a/^,  which,  as  we  shall 
see,  measures  the  entire  effect  of  the  mortar  when  the  ulti- 
mate i-esistance  to  extension  is  supposed  to  be  equal  to  the 
ultimate  I'esistance  to  comj^ression,  is  identical  in  value 
with  the  expression  ^c'd^^  which  measures  the  effect  of 
mortar  when  the  resistance  to  compression  is  infinite,  and 
the  neutral  axis  at  the  edge  of  the  joint.  But  c  as  determ- 
ined on  the  first  supposition,  will  be  twice  as  great  as  o' 
determined  on  the  last. 

Let  us  suppose,  then,  the  ultimate  resistance  of  mortar 
to  extension  and  compression  to  be  the  same.  Let  c  repre- 
sent that  resistance.  When  the  joint  m  n,  fig.  4,  has  so  far 
opened  that  the  mortar  at  n  is  about  to  separate,  the  adhe- 
sion or  eftective  force  then  applied  at  that  point  will  be  c 
— the  ultimate  strength  of  the  cement.  At  the  same  in- 
stant the  mortar  will  be  compressed  at  m  to  the  full  extent 
<jf  its  capacity  to  resist ;  and  there  will  be  a  neutral  axis 
at  the  middle  of  the  joint,  where  the  mortar  is  neither 
extended  nor  compressed. 

Tlie  resistance,  X,  to  extension,  is  proportional  to  the 
extension,  and  increases  uniformly  from  the  middle  of  the 
joint,  M'here  it  is  nothing,  to  the  extrados,  where  it  is  c. 
It  is  given,  therefore,  at  any  point  at  the  distance  x  from 
the  neutral  iixis,  by  the  proportion 

jrd'  :  c  ::  X  :  A'=yj,  ;  (jl'^zmn^ 

The  elementary  force  is  therefore  -^.dx,  and  the  entire 

.  K 

resistance  to  extension — 


ROTATION.  207 

In  like  manner  we  find  tlie  resistance  to  compression  to 
be, 

When  we  refer  these  two  forces,  which  act  in  opposite 
directions,  to  any  center  of  moments  not  between  their 
respective  resultants,  we  must  regard  them  as  having  con- 
trary signs.  Referred  to  m,  the  common  center  to  which 
all  the  forces  of  the  system  are  referred,  the  force  J?, 
tending  to  prevent  rotation,  is  negative ;  the  force  R', 
tending  to  cause  rotation,  is  positive  ;  the  lever  arm  of  R 
is  ^mn=i\d' ;  the  lever  arm  of  R'  is  ^?nn=\d'.  Their 
combined  moment,  tending  to  prevent  the  rotation  of  the 
voussoir  a  h  m  n  r  <?,  around  the  point  m^  is 

rd' 

-(icr-i<r)=icYr>^ 

In  like  manner,  we  find,  at  the  vertical  joint,  above  the 

neutral  axis,  a  resistance  to  compression,  — ,  acting  with 

the  lever  arm  (y — !-<:/),  tending  to  prevent  rotation  around 

the  center,  m ;  and  below  the  neutral  axis,  the  force  — ^ 

acting  with  the  lever  arm  (y—id),  tending  to  cause  rota- 
tion around  the  point  w.  The  combined  moment  opposed 
to  rotation  is 

rd 

The  two  joints,  therefore,  oppose  to  rotation  the  moments 

\c(d'^+d'^). 

We  might  have  shortened  the  demonstration  by  assum- 
ing at  once  this  obvious  principle,  that  the  algebraic  sum 


208  THEORY    OF    THE    ARCH. 

of  the  moments  of  any  two  equal,  parallel,  and  opposed 
forces,  is  constant  for  all  centers  of  rotation  in  the  same 
])lane,  and  always  equal  to  either  force  nniltiplied  ])y  the 
distance  hetween  their  respective  directions ;  a  principle 
which  miglit  have  a  very  extensive  application  in  calculat- 
inir  the  total  effect  of  certain  coniLinations  of  timber  in 
wooden  bridges  and  other  structures. 

The  French  eno^ineers,  resfardins:  the  neutral  axis  as  at 
the  edge  of  the  joint,  give,  as  the  moment  of  resistance 
due  to  mortar,  ^cd^. 

We  have  Ijeen  the  more  particular  in  the  above  rea- 
soning, l^ecause  we  differ  from  Poncelet  in  estimating  the 
effect  of  mortar  upon  the  vertical  joint.* 


*  Poncelet  sa\-s,  pnge  201,  vol.  12,  Memorial  de  I'Officier  du  Genie,  after  giving 

as  the  effect  of  mortar,  — .■' : — ;  in  which,  f\<r.    4,   d'=mn;    d=ab  ; 

■Sy  3{H+h) 

y=.am  ;  h  =  EE' ;  H=zaC ;  c=:the  force  of  adhesion  upon  a  unit  of  surface, — 

"  We  are  led  inevitably  to  this  result,  by  the  principle  of  virtual  velocities,  in  con- 
sidering, at  the  same  time,  the  cohesion  on  the  joint  of  rupture  of  the  reins,  and  on 
that  of  the  ke\-,  and  in  supposing,  after  Mariotte  and  Leibnitz,  that  tliis  force  is 
proportional  to  its  distance  from  the  point  of  rotation  of  each  joint.  The  justifica- 
tion of  tliis  result  presents  no  difficulty  except  in  that  part  which  relates  to  the 
second  of  these  joints,  which  is  not  usually  considered.     Now,  we  must  ol)serve  that 

the  resultant  of  the  elementary  forces  — {ii,  which  act  along  ah,  in  opposition  to  a 

d 

movement  of  rotation  at  a,  and  of  wliieh  tlie  moment,  in  reference  to  that  point,  is 
— —dx=:\cd^,  can,  as  to  this  movement,  be  replaced  by  a  horizontal  force 

— ,  supposed  to  be  apiiliod  at  the  edge  of  the  intrados,  m,  of  the  lower  joint;  and 

as  the  latter  is  opposed,  at  the  same  time,  to  the  action  of  the  horizontal  thrust,  F, 
in  ite  tendency  to  overturn  the  entire  system  of  the  semi-arch  and  its  pier  around  the 
exterior  edge  of  the  base  of  t!ie  latter,  witii  a  lever  arm,  //+// — y,  whilst  that  of 
/•'is  ll-k-h,  we  see  that,  in  relation  to  this  movement,  it  must,  in  its  turn,  be  replaced 

by  a  horizontal  force  cd— ^  acting  at  a,  to  resist  the  movement  in  question, 

3.v(//-t-A)  °  4  . 

conjointly  witli  llie  force  which  is  capable  of  overcoming  the  cohesion,  on  th.'  lower 

joint  m  71,  with  the  lever  arm  ;/,  and  of  wliich  the  known  expression  is  ^ — •'     We 

ay 

have,  therefore,  as  the  total  resistance,  the  quantity, 

cd'(IlJrh—y)     cd'^  _c{d?-^d")  cd" 


:iy{JI+k)  3y  3y  '6[H  +  /,)' 


UNIT     OF    WEIGHT.  209 

16.  We  must  caution  the  reader  tliat  the  unit  of  weight 
iu  the  foregoing  equations,  is  tlie  weight  of  a  culjic  unit  of 
masomy.  To  make  the  terms  involving  c  homogeneous 
with  the  other  terms,  we  must  regard  c  as  the  ratio  of 
the  strength  of  adhesion,  in  pounds,  upon  the  unit  of  su:- 
face,  to  the  weight  of  the  same  unit  cubed. 

For  instance,  suppose  the  force  required  to  separate  a 
square  foot  of  mortar  to  be  3,000  pounds,  the  cubic  foot  of 
masonry  weighing  150  pounds,  we  must  have  c^YtV  =-^- 

17.  Collecting  the  above  formulae,  we  have,  as  the  gen- 


wlucli  makes  the  position  of  the  joint  of  rupture  at  the  reins  depend  upon  the  height 
of  the  arch  and  that  of  the  pier." 

In  the  above  transhation  we  have  changed  the  notation,  to  suit  our  diagram  and 
text. 

This  very  distinguished  philosoplier  has,  we  think,  here  fallen  into  an  error. 

The  force  — ,  which  the  mortar  opposes  to  rupture  at  the  crown  of  the  arch, 

and  of  which  the  moment,  in  relation  to  the  upper  edge  of  the  vertical  joint,  is  _- 

is  not  opposed  to  rotation  around  m,  the  inner  edge  of  the  joint  of  the  reins.  Its 
whole  tendency  is  to  cause  such  rotation.  The  equal  and  opposite  force,  or  resist- 
ance to  compression,  acting  at  a,  the  summit  of  the  crown,  is  alone  opposed  to 

rotation.  The  point  of  application  of  the  force  — ,  or  resistance  to  extension,  is  situ- 
ated upon  the  vertical  joint  at  two-thirds  its  length  from  the  extrados.  Its  lever 
arm,  in  reference  to  m,  is  {y  —  \d) ;  and  the  lever  arm  of  the  equal  and  opposite 
force,  or  resistance  to  compression,  acting  at  the  extrados  of  the  crown,  is  y.  The 
combined  moment,  in  relation  to  m,  is  therefore 

Wherever  we  suppose  the  neutral  axis  to  be,  the  total  resistance  to  compression 
must  necessarily  be  equal  to  the  whole  resistance  to  extension ;  and  this  does  not 
cease  to  be  true  even  if  we  suppose  one  of  these  resistances,  per  unit  of  surface,  to  be 
infinite, — that  is,  if  the  neutral  axis  be  at  the  edge  of  the  joint. 

M.  Poncelet  seems  to  have  fallen,  inadvertently,  into  one  or  two  other  errors, 
which  make  bis  final  result  nearly  correct. 

The  force  — ,  which  we  can  substitute  for  the  "  resultant  of  the  elementary  forces 

^y 

^dx,"or—-,  in  relation  to  the  opening  of  the  joint  of  the  crown,  for  that  very  rea- 
d  2 

son  we  can  not  so  substitute  in  relation  to  any  other  center  of  rotation. 

Finally,  we  are  not  at  liberty  to  refer  the  moments  of  a  system  in  equilibrium  to 
different  centers. 


IJIO  THEORY     OF    THE     ARCH. 

eral   \nliie  of  tlie  liorizontal  thrust,  in  tlie  common  mode 
of  rupture, 

AVitliout  suroli.  ^    l' tlio  inaxi- ^  „ 

or  a<lhesion  of  r  -    iimin  val-  >-  F'=-L  .  (1) 

iimrtar,  )    (    iic  of         )  ^ 

AVitl.  surcharge,  do  j^,^Sr+Sp+_S"p"+.. 

y  ^'' 

AVith  a.lliesiou  .  vi _^P      ^c{d^-\-d!^)  ,o\ 

of  mortar,  '  ~~y      ^         y 

AVitl.both,  do  F'^Sp^S-p^S'p^^  dl^^^^^^ 

y  y 

Ami,  for  the  thickness  of  jiier,  ii^  representing  the  thrust, 
however  determined, 

AVitliout  adhesion  of  mortar  at  )  ^      r.     ,      ,^  /_> 

tl.o  W.O  nf  th.  nior  r^^Fx  l  =  M  (o) 


the  base  of  tlie  pier,  ) 

Tith   adhesion  of  mc 
the  base  of  the  pier, 

e  is  the  tliickness  of  pier,  8  the  coefficient  of  stability. 


AVith   adhesion  of  mortar  at  )  ^  r;      ^r  ,  ,    i  tn\ 


18.  No  arch  could  stand  in  bare  equilibrium.  Experi- 
ence has  shown  that  the  semi-arch  and  pier  must  have  a 
certain  excess  of  stability  over  the  thrust ;  and  the  French 
engineers  liave  provided  for  this  excess,  in  the  equations 
which  detei-mine  the  tliickness  of  pier,  by  assigning  to  the 
tlirust  an  increased  value — multiplying  it  by  the  coefficient 
of  ,'st((hiJifij^  generally  assumed  at  l.OO  or  2  in  the  heavy 
arches  of  fortifications,  l»ut  wliitli  may  be  safely  assumed 
at  less  in  establishing  the  piers  of  ordinary  bridges.  We 
shall  give  a  discussion  of  tliis  important  subject  hereafter. 

19.  Tlic  tlirust  at  the  crown,  due  to  the  mutual  action 
of  the  semi-arches,  though  not  sufficient  to  turn  over  the 
semi-ai-ch  and  ]>icr,  may  under  some  circumstances  cause 
the  whole  mass  to  slide  (»utwardly  on  the  biise  of  the  pier. 
The  e(piation  of  equilibrium,  calling  W  the  whole  surface 
of  the  semi-arch  .nid  jmci-,  and  /  the  friction  or  ratio  of 


SECOND     MODE     OF    RUPTURE-SLIDING.  211 

resistance  to  pressure,  will  ohvioiisly  be,  ^=  Wxf ,'  and 
the  practical  formula,  calling  6'  tlie  coefficient  of  staljilit}', 
will  be  ^'F=  Wxf. 

If  we  wish,  however,  to  take  account  of  the  adhesion 
of  mortar  upon  the  base  of  the  pier,  we  shall  have 

b'F=Wxf-\-cXe.  (7) 

As  to  this  particular  danger,  the  weakest  joint  is  gen- 
erally at  the  springing  line,  or  very  near  it.  Applied  to 
that  case,  W  would,  of  course,  represent  the  surface  of  the 
semi-arch,  and  e  its  thickness  at  the  springing  line. 

If  the  masonry  be  well  constructed,  even  of  poor  mor- 
tar,— that  is,  well  bonded  together, — no  sliding  can  ever 
take  place  at  the  springing  line  or  base  of  the  pier. 

The  thickness  of  pier  should  always  be  determined  in 
view  of  rotation  alone.  In  those  very  rare  cases  in  Avliich 
any  danger  of  sliding  can  still  remain,  care  must  be  taken 
in  the  preparation  of  the  foundations  to  render  such  mo- 
tion impossible.  The  most  economical  and  effective  expe- 
dient is,  probably,  that  of  giving  to  the  base  of  the  pier  a 
slight  inclination — say  one  foot  in  ten. 

SECOjS^D    ]\rODE    OF    RUPTURE — SLIDING. 

20.  When  the  thickness  of  an  arch  compared  with  its 
span  is  very  great,  the  horizontal  thrust  no  longer  arises 
from  any  tendency  to  rotation,  the  lever  arm  ^>,  ecpiatiou 
(1),  being  very  small,  or  even  negative. 

The  thrust  arises  from  the  tendency  of  the  upper  vous- 
soirs  to  slide  down  their  beds  or  joints.  It  is  precisely  like 
the  thrust  of  an  embankment  of  earth,  and  is  determined 
in  the  same  manner,  viz.,  hj  the  "  prism  of  maxinium 
thrust." 

Let  /represent  the  friction  along  the  joint  m  n,  fig.  5. 

"     a         "         the  angle  of  fiiction,  measured  from  a 
horizontal. 


212  TnEORY    OF    THE    ARCH. 

Let  V  represent  the  angle  between  the  joint  //i  //,  and  a 
vertical  line. 

"     S        "         the  surface  of  the  segment  ((  h  m  n  a. 

'•  P'  "  tlie  horizontal  force  acting  on  the  joint 
a  h^  necessary  to  keep  the  segment 
from  sliding  down  the  joint  m  n. 

We  have,  perpendicular  to  tlie  joint  m  ??,  the  force 
P' y.  COS.  i'+zS'sin.  y  ;  and,  parallel  to  the  joint  m  n^  the 
force  /S'cos.  y— P' sin.  V  ;  hence  the  erpiation  of  eq^uilib- 
rium,  [P'  COS.  r+zS'sin.  t')/=:iS'cos.  v—P'  sin.  v. 

Taking  account  of  the  adhesion  of  mortar  along  the 
joint  mn=:d\  we  have, 

(P'  COS.  u-f-/S'sin.  v)f-\-cxd'=S COS.  v—P'  sin.  v. 

We  have,  therefore,  as  the  general  expression  of  the 
horizontal  thrust, 

Without  [   j  the  maximu.n  }  ^■^^o'^- ^'-Ai!J:iJ^^^eotang.(«  +  .);  (8) 
adhesion,)    (       vahie  ut       j  sin.w+/cos.v 

a.lhesion,  f    (  \  ^  \         J     sin.(a  +  <')  ^  ^ 

As  in  the  case  of  rotation,  we  must  suppose  the  angle 
V  to  vary  from  the  summit  towards  the  sjn'inging  line,  and 
ascertain  the  maximum  value  of  P  in  (8)  or  (9).  This 
value,  which  is  the  sUdiiKj  thrus-t  of  the  arch,  we  shall 
desi<?nate  1>v  P. 

21.  E(iuations  (8),  (9),  include  the  case  of  a  surcharge, 
if  we  su])j>ose  Sto  take  in  all  the  load  wliicli  rests  verti- 
cally iijxm  the  segment  a  h  in  n. 

22.  Ill  arolies  of  common  use,  tliis  mode  of  rui)ture  can 
never  take  place.  The  resistance  necessary  to  prevent 
rotation,  wliich  one  lialf  of  the  stable  arch  necessarily 
opposes  to  tlie  <»tli('i'  lialf,  is,  in  almost  all  practical  cases. 


THIRD     MODE    OF     IIUI'TURE— ROTATION  213 

greater  tlian  tlie  force  necessary  to  prevent  sliding.  If  it 
were  otlierwise,  if  P  were  greater  than  F,  the  former 
would  l)e  the  pro|)er  thrust  of  the  arch,  and  would  take 
the  place  of  j?j^in  equations  (5),  (G),  (7),  arts.  17,  19,  wlien 
we  wisli  to  determine  the  thickness  of  j)ier. 

23.  We  commit  an  error  in  favor  of  stal)ility,  in  sup- 
posing tbe  force  P  to  l)e  applied  at  a^  tlie  extrados  at  tlie 
crown.  Its  true  point  of  application  is  always  intermedi- 
ate between  a  and  Z»,  and  generally  near  the  middle  of  that 
joint. 

24.  The  effect  of  adhesion  on  any  joint,  in  resisting  a 
force  acting  parallel  to  that  joint,  we  have,  in  the  usual 
manner,  measured  by  the  product  of  the  force  of  adhesion 
upon  a  unit  of  surface  into  the  length  of  the  joint  expressed 
in  the  same  unit. 

It  does  not  follow  that  the  force  c^  which  will  separate 
a  square  foot  of  cement  when  acting  at  right  angles  to  the 
joint,  will  tear  asunder  neither  more  nor  less  when  acting 
parallel  to  the  joint. 

The  rule,  however,  is  the  only  one  that  lias  been 
offered,  and  it  is,  without  doubt,  sufficiently  correct. 

It  would  perhaps  be  better,  proceeding  expei-imentally, 
to  regard  the  adhesion  solely  as  giving  a  greater  value  to 
the  friction. 


THIRD    MODE    OF    RUPTURE ROTATION. 

25.  This  is  still  more  uncommon  than  the  second.       *" 
Gothic  arches,  fig.  6,  and  arches  very  light,  and  lightly 

loaded  at  the  crown,  and  overloaded  at  the  reins  fig.  7,  are 

liable  to  this  mode  of  rupture. 

As  compared  with  the  usual  mode  of  rupture,  arts,  -l, 

5,  figs.  2,  3,  every  thing  is  reversed.     The  crown  lises ;  the 

reins  ftdl  in  ;  the  vertical  joint  opens  at  the  extrados,  the 


214  THEORY     OF    THE     ARCH. 

adjacent  segments  toncliing  only  at  the  intrados  ;  the  rehis 
oj^en  at  tlie  intrados,  the  adjacent  segments  touching  only 
at  the  extrados  ;  the  arch  still  divides  into  four  pieces  ;  the 
upper  segments  turn  outwardly  on  the  exterior  edges  of 
the  joints  at  the  reins  ;  the  lower  segments  turn  inwardly 
on  the  interior  edges  of  the  lower  joints. 

Tlie  active  force  which  pushes  over  the  upper  seg- 
ments, acts  in  this  case  at  the  intrados  of  the  crown.  It  is 
generated  entirely  hy  the  effort  or  tendency  of  the  semi- 
arch,  or  some  segment  of  the  arch  alcove  the  s})ringing 
line,  to  revolve  under  its  own  weight,  turning  on  the  inner 
edge  of  its  lowest  joint.  To  obtain  the  exact  value  of  this 
thrust,  we  must  determine  the  maximum  value  of  tlie  ex- 
pression  -4 ,  fig.  4,  differing  from  the  horizontal  thrust  in 

the  first  mode  of  rupture  only  in  the  lever  arm  of  the 
thrust,  which  is  now  hn  ■=zy\  instead  oi  am=:y. 

Now,  the  horizontal  force  at  ^,  fig.  4,  necessary  to  cause 
any  other  segment  to  commence  rising  at  the  crown,  turn- 
ing outwardly  upon  t?,  is  ^,  ?/"  representing  the  lever 

arm  l>n\  and  g'  the  distance  of  ??,  the  center  of  rotation, 
from  the  vertical,  (/g\  dropped  from  the  center  of  gravity  of 
the  segment.  If  any  segment  he  caused  to  rise,  it  will  ob- 
viously l»e  that  wliicli  offers  the  least  resistance.     We  must 

therefore  by  ti-i:d  find  the  iiiinimvrn  value  of  — ^.     If  the 

V 
hoiizoiital  thrust,  as  defined  above,  or  rather  if  the  ma^i- 

nniiii  vain*'  of  ^ ,  which  is  a  little  greater  tlian  tliat  tlirust, 

exceed  this  minimum  valuer,  tlie  arch  will  fall.  This  may 
take  j)lace  in  very  light  circular  or  segmental  arches,  sur- 
charged hf^rizontally  ;  as  will  be  proved  hereafter. 


FOURTH     MODE    OF     RUPTURE— SLIDING.  215 

FOURTH   ]\rODE    OF    RUPTURE SLIDING. 

26.  The  thrust  at  the  key,  generated  by  the  niutnal 
action  of  the  semi-arches,  miglit,  it  would  seem,  cause  tlie 
arch  to  slide  outwardly  upon  some  of  its  lower  joints.  The 
horizontal  force  at  the  key,  necessary  to  cause  such  move- 
ment is  ScotsLng.(^v—a)  :  for  notation,  see  art.  20.  Its 
least  value,  in  practice,  is  always  at  the  springing  line, 
where,  if  anywhere,  sliding  will  take  place. 

The  equation  of  equilibrium  of  the  semi-arch  upon  its 
springing  line,  supposed  to  be  horizontal,  has  already  been 
given,  art.  19. 


SECTION  II. 


27.  We  have  given  in  the  first  section,  the  theory  of 
the  arch  in  its  most  general  terms. 

We  shall  hereafter  give  geometrical  methods  of  deter- 
mining the  thrust  at  the  crown,  and  all  the  other  elements 
of  any  case  that  is  likely  to  arisQ,  in  terms  ecpially  general ; 
that  is,  independently  of  the  pai'ticular  nature  of  the 
curves  of  the  extrados  and  intrados. 

As  circular  arches  are  by  tiir  the  most  common,  and 
admit  of  the  most  precise  calculations,  w^e  shall  first  apply 
the  theory  to  them. 

The  tables  calculated  by  M.  Petit,  Capitaine  du  genie, 
and  the  more  extensive  tables  prepared  for  this  work  and 
now  published  for  the  first  time,  will  enable  us  to  dis]:)ense 
with  calculations  in  many  cases,  and  greatly  abridge  them 
in  nearly  all. 

The  brevity  and  simplicity  which  characterize  the  gen- 
eral expressions  of  the  horizontal  thrust,  unfortunately 
vanish  when  we  submit  the  most  simple  cases  to  calcul- 
ation. 

3 


21^  THEORY     OF    THE    AUCII. 

CIRCULAR    ARCHES,    INTHADOS    A>'D    EXTRADOS    PARALLEL. 

•28.  lu^Vning  to  fig,  8,  let  R  l)e  the  radius  of  tlie  ex- 
tradns ;  /•  tliat  <>t*  the  iutiados  ;  v  tlie  are  "\vliich,  in  the  oii'cle 
\vli<  )sr  radiiH  is  unity,  measures  the  angle  of  any  joint  m  n 
with  a  vertical. 


Kesume  equation  (1),  F'^'^.  "We  have,"  >S  =  ir(i?'- 
ri^«-r«)(l-cos.  ^0  ^ 
/-)  ;  p=r  SHL  v-i^ '^ir~^^~) ^^  y=^-''  ^^^-  '^  / 

These  values  substituted  give 

Sp_Sr(I^-r')v  sin.  v-2(B'-7-')(l  -cos,  v)  .     ^^^^ 
~  y~  6(it— 7'cos. -t;) 

an  expression  of  the  horizontal  force  F'  Avhich,  applied  to 

the  extrados  of  the  key,  can  prevent  the  voussoir  a  h  in  n  a, 

coiresponding  to  the  angle  ^',  from  turning   round    the 

point  m. 

Tliis  we  can  simplify  by  introducing  the  ratio  of  the 
p 

t\v(»  radii,  7^=     ,  which  crives 

'  /•'  ° 

JC— COS.  V  y  \       J 

The  value  of  /',  or  the  inclination  of  the  joint  of  rup- 

"  Distance  from  the  conter  (if  the  c-irclc,  Hg.  8,  on  the  bisecting  line  Cff,  of  the 

,  ,  ^  ./2x /?2  sin.  Aw     .^2  sin.  An 

cent^Tof  giavit\-of  the  sector  Can  =  l —=IR —=d 

vli  V 

..,./,;  s  2  sin.  i»/ 

V 

surf,  r  a  n  x  d — surf.  C  b  vi  x  (F 

nng  II  h  III  >i  a:= •    , — -,- . ' 

surf,  of  nng. 

_  (/?*— r»)26in.^« 

p=>iiff'=iiim' — jr'wjzrr  sin.  v — Off  x  sin.  iv=:. .  ns  nbovc 
Surface  Ca  ti  =  ^vli'';  2  sin.'iv  =  l — cos.  v. 


CIRCULAR    ARCHES.  217 

tiire  wliicli  corresponds  to  the  maximum  value  of  F\  may 
be  ol)taine(l  in  the  usual  way  by  differentiation.  Tlie  nu- 
merator placed  equal  to  zero,  gives,  by  reduction, 

^  X^  —  1 

COS.  v^{\—K cos.  v)-. =  K—  4  , ,     -  :  f  1 2 ~) 

and  the  corresponding  value  of  F'  is 

i^=.=(j-(A'^-l)(l+^:j£^cos.^)-J(^-»-l))    (13) 

These  expressions  establish  the  highly  important  gen- 
eralizations that,  in  all  similar  arches  having  the  ratio  K 
of  the  two  radii  constant, — • 

1st.  The  thrust  is  proportional  to  the  square  of  the 
radius  of  the  intrados,  or  to  the  square  of  any  other  linear 
dimension  ; 

2d.  The  angle  of  rupture  is  constant. 

Let  11  represent  the  numerator  of  the  second  member 
of  equation  (11). 

Let  z  represent  the  denominator  of  the  second  member 
of  equation  (11). 

rti 

The  condition  that  i^',  =  -,  shall  be  a  maximum,  gives 
zdu  —  udzz=i^  ;  or  -=:y-.     In  this  way  (13)  was  obtained. 

Z       (J/Z 

The  value  of  v  obtained  by  trial  from  (12)  and  substi- 
tuted in  (13),  gives  the  true  thrust.  In  this  way,  M.  Petit 
has  calculated  table  A^  giving,  either  directly  or  by  pro- 
portional parts,  the  horizontal  thrust  and  the  angle  of  rup- 
ture for  all  values  of  K ;  that  is,  for  all  semi-circular  arches 
which  have  a  constant  thickness,  or  the  intrados  and  extra- 
dos  parallel. 

29.  To  illustrate  the  march  of  F'  througli  tlie  (piad- 
rant,  as  we  assign  to  K  a  particular  value,  and  to  v  in 
equation  (11)  a  succession  of  values,  we  have  calculated 


218 


THEORY    OF    THE    ARCH. 


the  folluwinjT  taLle,  of  Avhicli  the  hist  column,  to  make  the 
result?*  more  taiipfihle,  gives  the  values  of  ^'  in  pounds ;  r 
l>eiug  taken  at  10  feet,  and  each  unit  multiplied  hy  150, 
the  assumed  weight  of  a  tiihic  foot  of  masonry.  /r=1.20 ; 
/— 10  feet;  11=11  feet. 


V  — 

rlO° 

F' 

=  r'x. 013855  =  ] 

lOxlOx 

150  X 

.013855=  208.00 

pounds, 

20° 

t( 

r«x. 044078  = 

15000 

X 

.044078=  070.00 

u 

30° 

t( 

r'x. 075111  = 

15000 

X 

.075111  =  1127.00 

u 

40° 

(( 

r«x. 090008  = 

15000 

X 

.090008  =  1450.00 

(( 

50° 

ti 

r'x. 108370  = 

15000 

X 

.108370  =  1025.55 

u 

55° 

t( 

r'x. 110980  = 

15000 

X 

.110980  =  1064.70 

1( 

5V° 

u 

r»x. 111470  = 

15000 

X 

.111470  =  1672.05 

(( 

60° 

u 

r*x.lll700  = 

15000 

X 

.111700  =  1675.50 

(( 

03° 

(( 

r«x. 111390  = 

15000 

X 

.111390  =  1670.85 

u 

05° 

u 

r*x. 110740  = 

15000 

X 

.110746  =  1661.19 

u 

70° 

u 

r*x. 108100  = 

15000 

X 

.108160  =  1662.00 

ii 

80° 

i( 

r'x. 099300  = 

15000 

X 

.099360  =  1490.00 

u 

90° 

(( 

r-x. 085600  = 

15000 

X 

.085660  =  1285.00 

u 

The  an<2:le  of  maximum  thrust  is  about  60"  and  near 
that  angle  the  variations  of  F'  are  very  small, — only  53^- 
pounds  from  50°  to  70".  The  exact  value  of  the  angle  of 
I'upture  or  angle  of  maximum  thrust,  is  a  matter  of  no 
importance. 

30.  If  we  suppose  r=0  in  equation  (10),  the  thrust 
becomes  nothing.  The  same  supposition,  ^'=0,  in  (12) 
gives. 

from  wliieh  we 'deduce  two  j)ositive  roots, 
K=\  ;  and,  7l"=2.732 

For  these  two  values  of /r,  there  is  no  thrust,  and  no 

angle  of  rupture,  except  that  of  the  key. 

p 
The  first  value,  7v^=    -  =  1,  corresponds  to  an  arch  in- 
finitely tliin,  witliout  weight,  and  therefore  without  thrust. 


EFFECT    OF    MORTAR.  219 

The  second  value,  K= 2.732,  corresponds  to  arches  of 
great  thickness,  in  which,  however  small  we  make  the 
angle  v^  there  can  be  no  thrust,  because  the  center  of 
gravity  of  the  segment  resting  on  any  joint,  will  fall  within 
the  intrados  of  that  joint. 

For  all  values  of  If  greater  than  2.T32,  the  thrust 
would  be  negative ;  that  is,  it  would  require  a  positive 
force  to  turn  over  any  voussoir  if  one  half  the  arch  were  to 
stand  by  itself.  We  are  here  speaking  of  rotation.  Such 
arches  have  a  very  decided  thrust  from  the  tendency  of 
the  upper  voussoirs  to  slide  uj)on  their  beds. 


EFFECT   OF   MORTAR. 

31.  Let  us  now  resume  equation  (3). 

^._Sp   ,(<r-+<P) 
y    """     y      ' 

an  expression  of  which  the  maximum  value  will  give  the 
thrust  as  modified  or  diminished  by  the  adhesion  of 
mortar. 

WehnYe d=cr =li—)'=)'(ir— I);  and7/=r(^— cos.r); 

6y  A  — COS.  y 

To  illustrate  the  effect  of  the  mortar,  let  us  asrain  take 
up  the  arch  corresponding  to  A^=:1.20,  art.  29.  Let  us 
sui^pose  the  adhesion  of  mortar  to  be  3000  pounds  per 
square  foot,  and  the  weight  of  a  cubic  foot  of  masonry  150 
pounds ; 

.   .  3000  T       -,     . 

2:ivm2:,  art.  16,  c=  ——-  =  20;  and  reducm": 

(K-lf  ^      0.8  1 


IC—cos.v  3       1.20— cos. -i; 


OOQ  THEORY     OK    THE    ARCH. 

Eeeapitnlatiiii,^  tlie  tal)le  given  in  art.  20,  and  snl)tract- 
ini,^  tVoni  each  value  of  F'  tlie  eftect  of  mortar  correspond- 
ins:  to  the  same  value  of  v.  we  have — 

Pounds.  Pouinls.  Pounds. 

r  =  10°  i^'  =  r'x.013855  —  rx  1.2392=  208.00  — 1651.00=  — 1443.00 
».  20°  "  7-«x.044G78  —  rx  1.0244=  670.00  —  1537.00  =  —  867.00 
u  30O  u  r' X. 075111— rx  .8000  =  1125.00  —  1200.00  =  —  75.00 
u  40°  «  r' X. 096668  — rx  .6145  =  1450.00—  922.00=+  528.00 
-'  50°  "  r''x.l08370  — ?-x  .4800  =  1625.55—  720.00=+  905.55 
"  60°  "  r*x.lll700— rx  .3810  =  1675.50—  571.50=  1104.00 
"  70°  "  r*  X. 108160  — rx  .3110  =  1622.00—  467.00=  1155.00 
"  80°  "  r' X. 099360  — ?-x  .2600  =  1490.00—  390.00=  1100.00 
"     90°    "       r^  X. 085660  — rx    .2200  =  1285.00—   330.00=        955.00 

The  true  thrust  or  greatest  vahie  of  F'  corresponds 
now,  we  see,  to  about  70",  and  is  reduced  from  1G75  pounds 
to  1155  pounds. 

32.  We  simplify  the  calculation,  without  any  sensible 
error,  by  supposing  ?',  in  the  term  involving  the  adhesion 
of  mortar,  equal  to  60",  which,  in  general,  diffei*s  but  little 
from  the  angle  of  maximum  thrust.  This  gives  cos.  v=.\. 
We  shall  therefore  be  aT)le,  with  little  labor,  to  correct  the 
thrust,  as  given  by  the  4tli  column  of  table  A,  or  obtained 
])V  a  direct  calculation,  when  we  assign  to  the  adhesion  of 
mortar  a  particular  value. 

Let  C  represent  the  decimal  of  that  column,  or  the 
decimal  obtained  by  calculation, — such  that  7'-X  6' is  equal 
to  the  maximum  thrust  without  regard  to  mortar.  We 
have,  a^i  the  thrust  diminished  by  mortar, 

7?-=/^  6'-^  J^^  (U) 

The  numerical  factor,  |  ^^  t^  -=.C\  is  very  ea.sily  cal- 
culated when  we  kno\\'  tlie  value  of  K.  For  instance, 
/r=1.20,  gives  6''  =  .01905;  and  (14)  becomes,  substitut- 


EFFECT    OF    MORTAR.  221 

ing  for  C  tills  value,  and  for  C  the  value  given  by- 
table  A, — 

^-"=^•2  X  .1114— r  X  0  X  .01905, 
whicli  reduces,  when  c  =  20,  to        F^r'^  x  .1114 — r  x  .381, 

"  "  "      r  =  10  feet  to  i^=ll. 14  — 3.81  =  7.33  =  1100  pounds, 

differing  sliglitly  from  tlie  thrust  above  given  for  the  same 
case ;  our  calculation  not  giving  precisely  the  tabular  value 
of  the  thrust. 

33.  Equation  (14),  which  can  be  put  under  the  form 

F^-v^Y.  C—rX.cG\  leads  to  this  generalization. 

jp 

While  the  thrust  of  similar  arches  (A''= —  beinsf  coil- 

^  r  ^ 

stant),  independently  of  the  mortar,  increases  as  the  square 
of  the  radius  of  the  intrados,  the  effect  due  to  the  mortar 
increases  only  as  the  first  power  of  that  radius.  Conse- 
quently, in  arches  of  large  span,  the  effect  of  mortar  be- 
comes insensible  ;  and  in  arches  of  small  span,  this  effect 
may  reduce  the  thrust  to  nothing. 

Placing  the  second  member  of  (14)  equal  to  zero,  we 
have  at  once  the  radius  of  the  intrados  corresponding  to  an 
arch  without  thrust, 


r'C=%rc\-j;^-^  giving  r=0  ;  and  r  =  -  x  ^^-— -=c  x 


If  K=1.20,  we  have,  from  table  A,  6'=. 1114,  and  by 
calculation  6^' =.01905.  If  we  furthermore  suppose  C'=20, 
we  have  7'=3'.42. 

This  corresponds  to  an  arch  of  6'.84  span  and  a  little 
more  than  8  inches  thick. 

If  ^=1.50;  <:?=20,  we  have  as  the  radius  of  an  arch 
without  thrust,  r=:9'.66. 

If /ir=i2  ;  c=20,  we  have  r=:34'.14. 

The  general  principles  announced  above  relative  to  tlie 
effect  of  mortar  upon  similar  arches  of  different  spans,  are, 
it  is  evident,  of  universal  application,  whatever  be  the 
curves  of  the  extrados  and  intrados. 


222  TIIEORV     OF    THE    ARCH. 

EFFECT    OF    SURCHAKGE. 

34.  E(|nation  (i>) 

T'——LJ- — L—i —  =  ^-H — ^— — —.   ffives   tlie   force 

V  2/  _^  .'/  '  ^ 

F'   necessary   to  sustain  any  siii)])osed    se2:ment    of    tlie 

arcli  wlieii  the  extrados  is  loaded  witli  tlie  weights  >S",  /S"', 

ttc,  with  their  centers  of  gravity  in  vertical  lines  at  tlie 

distances  p\p\  etc.,  from  the  center  of  rotation,  or  ])oint 

m  of  the  intrados. 

If  the  surcharge  l)e  continuous,  and  nearly  constant  in 

vertical  depth,  find  in  taljle  A  the  value  of 

Sp       ori       w    +1-    ^^^  ^^'p'-\-S"p"+^^^ 
-rL^:=:^C\  and  to  this  add  ~^-^ ~ — 

y  Jt  —  ?'COS.v 

calculated  according  to  tlie  circumstances  of  the  case.  The 
sum  will  l)e,  with  sufficient  exactness,  the  thrust  increased 
by  the  surcharge. 

L(4  the  surcharge,  reduced  in  depth,  if  necessary,  to 
give  it  the  density  of  the  masonry,  l)e  of  the  constant  ver- 
tical depth  t.     We  shall  have — 

j)'  =  iiin.  v[r—\R)  =  7'  sill.  v[\ — ^A') ;  *S'  =  /-/A''sin  v  ; 

,  Sy     rtK{2-K)         sin.2/'  Til- 

and        -  —  — ^Xr^ ;  which  becomes  zero, 

//  2  jtL— COS.  V 

for  all  values  of  v,  when  7^=2,  or  11=2 r.  The  center 
of  gravity  of  the  surcharge  then  falls  upon  m. 

sill  ^  'V 

The  variable  fiictor,  --f^ ,  we  find  by  differentia- 

'  A  —COS.  V 

tion   to  be    a    maximum  when  C(,)s. t' =  A"— [  A"'^  —  1;  and 

sin.^  /;  ,  ,,       /  -.^,, — -\ 

A  —COS.  V       ^ 

AW-  ha\(',  tlicrd'oic,  witli  a  very  slight  error  in  favor 
of  stability,  for  the  thrust  increased  l)y  a  surchai-ge  of  uni- 
form depth, — 


EFFECT    OF    SUllCIIARGE.  223 

F=r'C^rtxK(2-K){K-\/lP^^)       (15) 

ill  wliicli  C  is  obtained  from  table  A,  for  tlie  given  value 
of  ^,  and  the  numerical  factor  K{^—K){K—\/jr^  —  \) 
=A^  is  easily  calculated  when  we  know  the  value  of  K. 

Example  ;  ir=1.20,  which  gives  iV=.5152.  Table  A 
gives  6'=. 1114;  hence  i<'=y^X.1114+r^X.5152  ;  sup- 
pose t—\r;  we  have  i^=7llll4+;ll030  =  ?l2144  ;  that 
is,  when  ^=1.20  the  thrust  is  nearly  doubled  l)y  a  sur- 
charere  of  uniform  thickness  /=4-r. 

*       2 

The  value  of  v,  which  renders  -^^     — —  a  maximum, 

7i  — cos. -y 

never  differs  more  than  four  degrees  from  the  value  of 

V  corresponding  to  the  same  value  of  ^in  table  A;  and 

it  is  the  property  of  a  maximum  to  exceed  but  little  the 

adjacent  values  of  the  same  function.     We  may  therefore 

regard  equation  (15)  as  exact. 

Table  F  gives  the  values  of  A^for  all  values  of  jST  be- 
tween 1.02  and  1.42 ;  these  values  being  the  same  in  all 
circular  arches  of  equal  thickness  throughout,  whatever 
load  they  may  sustain  in  addition  to  this  surcharge  of  con- 
stant depth. 

Taldes  A  and  F,  therefore,  give  the  thrust  required 
without  calculation.  See  discussion  and  use  of  those 
tables. 

35.  Let  us  take  up  another  variety  of  the  surcharge. 
Suppose  a  single  column,  represented  in  weight  by  a  sur- 
face one  unit  in  width  and  H  in  height,  to  rest  upon  the 

crown  ot  tiie  arch.      We  have,  1^  =i^--\—^ . 

y      A— COS. -y 

a,. 
The  maximum  value  of —^r^C*,  is  given,  without  re- 

y  . 

gard  to  surcharge,  by  table  A.     The  maximum  value  of 

-r^  '     '  "      obtained  by  the  calculus,  is    —====.      The 
X— cos.V  ^  '       f^IO  —  1 


224  THEORY    OF    THE    ARCH. 

trii.^  thriHt,  therefore,  with  some  error  in  excess,  is 

It  is  remarkal.lc  tliat  tliis  addition  to  the  thrust,  caused 
l.y  a  weight  upon  the  crown  of  the  arch,  is  independent 
of  ;■,  the  radius  of  the  intrados,  and  therefore  constant 
whih?  K  remains  the  same — that  is,  in  all  similar  arches. 
This  is  also  evident  from  general  considerations. 

Exami)le.  On  the  crown  of  an  arch,  20  feet  in  span 
and  2  feet  thick,  we  wish  to  throw  a  column  one  foot 
square  and  10  feet  high,  one  half  being  supported  by  each 

semi-arch. 

5 
"VVe  have  7^=1.20  ;   (7,  from  table  A,=:.11U  ;    ,-^^^- 

=  7.54;  7  =  /-.1114+«. 54  =  18.68. 

The  angle  i\  in  table  A,  corresponding  to  7^=1.20,  is 
59°.4:'. 

The  ande   v  which  renders  -w a  maxnnum,  is 

"=  7t  — COS.  t) 

\        T 
firiven  bv  the  relation  cos.  ?;=  y^=:-^.  =.83333  ;  or  i'=33°. 
®  *  K     It 

35'. 

The  true  thrust  would  correspond  to  tJ  =  about  50°. 

Tn  the  a1)0ve  example  we  commit  an  estimated  error  in 
excess,  or  in  favor  of  stability,  of  alxuit  five  per  cent. 
The  error  will  be  less  as  ^increases  or  7f  diminishes. 

The  effect  upon  the  thrust,  of  a  weight  placed  ujion  the 
crown  of  an  arch,  is  evidently  the  same  in  all  circular 
arches,  Avliether  otherwise  loaded  or  not. 


INTRADOS    AKD     EXTKADOS     PARALLEL.  '^O 


Zli> 


THRUST   OF    SEMI-CIECULAR    AECIIES,    SLIDING. INTRADOS 

AND    EXTRADOS    PARALLEL. 

36.  Resume  equation  (8),  art.  20.  V=SQoi.{a-\-v),  an 
expression  for  the  horizontal  force  re<iuired  to  prevent  the 
segment  whose  surface  is  S  from  sliding  doAMi  its  bed  or 
lower  joint  niii  fig.  5.     We  have 

jS=^o(Ji'^—r^)  ;  and,  substituting  lu-  for  7?, 

The  angle  of  friction  according  to  Boistard  is  3Y°.  14'; 
according  to  Rondelet,  30°.  Admitting  the  latter  value,  as 
more  favorable  to  stability,  we  have. 

P'=:|yX/i^2-l)Xt'Xcot.  (z'  +  30°)  (17) 

The  angle  v,  corresponding  to  the  maximum  value  of 
P\  deduced  by  trial  from  the  condition  h  sin.  2('?;+30°) 
=v,  or  directly  from  equation  (17),  is  a  little  over  26°,  and 
it  is  obviously  the  same  for  all  circular  arches  which  have 
the  extrados  and  intrados  parallel,  whatever  be  the  relative 
parts. 

Giving  to  v  this  value,  26°,  in  equation  (17),  we  have, 

F=:j'x(ir^-l)X.15^0-i;  (18) 

from  which  M.  Petit  has  calculated  the  slidinc:  thrusts  in 
table  A. 

It  will  be  seen  in  that  table,  of  which  a  full  discu?^siun 
will  be  given  hereafter,  that  the  thrust  due  to  rotation  is 
greater  than  that  due  to  sliding,  for  all  values  of  A"" 
from  0  to  1.44  ;  and  that  the  thrust  due  to  sliding  exceeds 
the  other  for  all  values  of  It  greater  than  1.44.  This 
value  of  ^  corresponds  to  an  arch  of  20  feet  span  and 
4'.40  thick  throughout. 


220  TUEORY  OF  THE  ARCH. 

EFFECT  OF  5I0RTAR. 

37.  Ixesiiine  equation  (9)  : — 

P'z=>S'eot.(./-j-r) — r— .— -— , 
^         ^     sin.  (<:/-[- ^7 

which  n(t^v  becomes 

o.r^o      ,x          w      ,      X      H-(7r-l)  COS.  30° 
F  ^^r\K  --1)6'  cot.  («+^') ^-^r^,-o^-' 

let  ns  suppose  7r=:1.50,  ;'=10',  c—1^  and  find  the  maxi- 
mum vahie  of  -P'. 

Pounds. 
r  =  20°  P'  =  r'x.l8.'306  — rx  1.1305  =  18.306 -11.305  =  '7.001  =1050 
u  26°"  r-x.l9130-rx  1.0444  =  19.130— 10.444  =  8.680  =  1302 
"  30°"  r-x.l8800  —  rx  1.0000  =  18.890  —  10.000  =  8.890  =  1333 
"     35°   "       ?•' X. 17803  — rx    .9555  =  1'7.800—    9.550  =  8.250  =  1237 

The  angle  i\  corresponding  to  the  maximum  value  of 
P\  is  now  about  30°.  The  effect  of  the  mortar  for  that 
anirle  is  rc(^K—\)  ;  and  it  is  nearly  the  same  for  i'=:2G°. 
"We  have,  therefore,  this  simple  formula  for  the  sliding 
tlirust  diminished  by  the  adhesion  of  moi-tar  : — 

p^r'C-rr{K-\);  (19) 

in  which  C  is  taken  directly  from  the  fifth  column  of  table 
A,  and  c  is  the  adhesion  of  niortai-  upon  a  unit  of  surface 
—art.  16. 

The  remarks  made  in  art.  33,  relative  to  the  eifect  of 
mortar  upon  similai-  arclics  of  different  spans,  are  equally 
api)lical)le  here.  The  second  meml)er  of  (19)  placed  equal 
to  zero,  gives,  as   the   radius   of  an  arch  witliout  sliding 

thrust,  /•=0,and  /-.^'iAzl). 

C 

For  f  =  2  ;  A'=1.50,  wliidi  irives,  tal.lc  .\,  r'=.191.10  ;  we  fiiul  r  =  5'.227  ; 
r..rf  =  2;  A"=1.20,     "  "  "  C=.06733;         "       r=5'.940. 

If  we  .suppose  c  =  20,  as  in   art.  33,  we  have,  as  the 


EFFECT    OF     SURCHARGE— SLIDIIsG.  00  7 

radius  of  an  arcli  witliont   sliding   tlirust,  for  /f^l.oO, 
r=52'.27  ;  for  7^=1.20,  ;■= 59'.4. 

The  effect  of  mortar  to  prevent  sliding  is,  we  see,  f\ir 
greater  than  its  power  to  resist  rotation.  In  tlie  latter 
case  it  lias  full  effect  only  at  the  outer  edges  of  the  open- 
ing joints,  and  its  influence  is  in  most  cases  still  further 
reduced  by  the  small  leverage  with  which  it  acts. 

38.  It  is  only  in  heavy  arches,  in  which  the  thickness 
is  nearly  half  the  radius  of  the  intrados,  that  the  effective 
thrust  is  determined  by  any  tendency  to  sliding. 

And  in  such  arches  we  can  not  rely  upon  any  adhesion 
of  joints,  unless  the  mortar  is  strongly  hydraulic,  and  con- 
siderable time  has  been  allowed  for  it  to  set  before  remov- 
ing the  centering. 


EFFECT    OF   SURCHARGE    UPON    THE   SLIDI^STG   THRUST. 

39.  The  general  equation  applicable  to  this  case  is 
P'=;S'cot.  (t,-j-30°)+.S"  cot.  (y  +  30°). 

We  can  always  suppose  the  vertical  depth  of  the  sur- 
charge, as  far,  say,  as  25°  or  30°  from  the  crown,  to  be  con- 
stant. Let  t  be  that  depth.  We  have  /S"=:it  sin.  t'X^ 
=  rtK^m.v,  and  8'  cot.  (i'  +  30°)  =  r^A^sin.  v  cot.  (t5-f-30°). 
The  maximum  value  of  this  expression  corresponds  very 
nearly  to  '^=25°;  while  the  maximum  value  of  /S"  cot.(y 
+30°)  corresponds  to  'y=26°. 

Adding  the  two  maxima  together,  we  have,  with  a 
very  slight  error  in  favor  of  stability,  the  sliding  thrust 
increased  ])y  a  surcharge  of  uniform  vertical  depth, — 

P^zr'C'+y'^/i^X  .29592.  (20) 

Cis  to  l>e  taken  from  the  fifth  column  of  table  A,  o])posite 
the  given  value  of  K. 


22S  THEORY     OF    THE    ARCH. 

The  numerical  coefficient  7l"x  .20592  is  given  in  the 
List  cohinin  of  tahle  F,  for  values  of  K  ranging  from  1.35 
to  1.50,  corresponding  to  arches  of  large  span  in  which 
the  slidins:  exceeds  the  rotation  tlirust. 

Tlie  tlirust,  increased  by  surcharge  and  diminished  by 
mortar,  is, 

P=r-6'+^'^A"X.20502-/r(7r-l),  (21) 

(7  being  taken,  as  Ijefore,  from  the  fifth  column  of  table  A. 

Exani[)le.     /r=2  ;  r;=2  ;  ^=5';  ;'=10';   weight  of  a 

cul)ic   foot   of  masonry  =  150  pounds.      Table  A  gives 

P=r'*X.459124-;'/x2x.29502-2/' 
=  45.912  +  29.592  — 2(>.=:.s:325  pounds. 

llc—o,  P=:11325  pounds. 

40.  Table  A  begins  witli  /r=  2.732,  above  whicli  the 
rotation  tlirust  is  less  tliaii  nothing.  The  sliding  thrust, 
however  goes  on  increasing  as  ^increases. 

Equation  (21),  expressed  more  generally,  becomes 

P=/-2x(/L'--l)x.l5304+/*^7rx.29592-/r(A-l)  (21)' 

in  whicli  ?•,  K^  t,  r,  may  have  any  values  whatever :  t  is 
the  mean  depth  of  the  surcharge,  whether  consisting  of 
one  or  more  masses ;  but  in  estimating  the  value  of  t,  we 
should  confine  our  attention  to  that  part  of  the  surcharge 
whicli  is  over  the  extrados  within  30°  of  the  summit. 

41.  In  the  investicfation  of  the  true  thrust,  as  modified 
by  surcharge  and  mortar,  three  distinct  cases  may  arise. 

I.  It  may  be  evidently  due  to  rotation.     See  art.  31,  and 

following. 
II.  It  may  evidently  be  due  to  sliding.     See  art.  30,  and 

fdlowini]^. 
III.  It  may  be  doubtful,  and  reipiire  an  investigation  of 
both  cases. 


THICKNESS     OF    PIER.  229 

The  greatest  of  the  forces,  F,  P,  required  respectively 
to  prevent  rotation  and  sliding,  will,  in  all  cases,  Ije  the 
true  thrust. 

The  third  or  doubtful  class  will  be  very  small  in  the 
hands  of  those  who  have  made  themselves  somewliat 
familiar  with  the  subject,  and  especially  with  the  tables 
contained  in  this  paper. 

THICKNESS    OF   PIER. 

42.  The  horizontal  thrust  at  the  key  enters  as  an  ele- 
ment into  two  questions :  1st,  the  thickness  of  the  i)ier ; 
2d,  the  ability  of  the  material  of  the  arch  to  stand  the 
pressui-e  at  the  summit  and  at  the  reins.  The  latter  ques- 
tion we  shall  take  up  hereafter. 

We  have  shown  how  to  obtain  this  thrust,  both  for 
rotation  and  sliding.  The  greatest  of  these  two,  which 
are  found  all  calculated  in  table  A,  is  the  true  thrust. 

We  defer  for  the  present  the  supposition  of  any  adhe- 
sion of  mortar,  or  of  a  surcharge. 

Let  i^"  represent  the  thrust ;  6' the  greatest  of  the  deci- 
mals in  columns  4,  5,  oj^posite  the  given  value  of  K.  We 
havei^=r2x6^ 

Let  h  represent  the  height  of  the  pier  from  the  base  to 
the  springing  line,  fig.  8  ;  /  the  lever  arm  of  the  thrust ;  e 
the  thickness  of  the  pier,  h  the  coefficient  of  stal^ility. 
We  have  Z=A-|-6^(;j^=A-f-Xr. 

We  must  give  to  the  pier  such  dimensions  that  its  mo- 
ment, increased  by  the  moment  of  the  semi-arch,  shall  be 
equal  to  the  moment  of  the  thrust  multi])lied  by  the  coeffi- 
cient of  stability. 

Let  n  =  \n{IO-\)',  m=l{K^-\y^ 
then    ?/r^=r  surface  of  semi-arch; 

^/iz-'^^r moment  of  semi-arch  in  reference  to  C ; 
(7i  — m)  ;'^=moment  of  semi-arch  in  reference  to  A  ; 
^>-^  X  6+ (?z— ;?/?.); '^=moment  of  semi-arch  in  reference  to  E 
or  E'. 


230  TIIEOUY     OF    THE    ARCH. 

We  liavt*,  therefore,  expanding  equation  (5), — 

\lLe-^nr'e+{n-m)r'=hCr\Kr-\-]i)  (;22) 

of  wliicli  tlie  solution  gives 

^=_;4VA/<J+2(567r+m-70;+266'      (23) 
r  n       \       ff  11 

Table  B,  calculated  l)y  M.  Petit,  gives  the  reduced 
form  of  this  equation  for  all  values  of  A"  between  1.10  and 
2,  on  the  supposition  of  strict  equilibrium,  ^=1 ;  the  value 
of  C  being  taken  in  each  case  from  table  A.  To  modify 
any  one  of  these  equations  by  introducing  the  coefficient  5, 
we  have  only  to  add,  under  the  radical,  to  the  coefficient 

of  -,  26^(5  —  1),  and  to  substitute  for  the  term  indepen- 

dent  of  -,  the  same  term  multiplied  by  the  coefficient  of 

stability. 

Example.     7r=1.50.     6',  being  always  the  greatest  of 
the  decimals  in  columns  4,  5,  is  .19130 

The  coefficient  of  y,  under  the  radical,  is  .19370 

To  this  we  must  add  267i(6-l)=  (5-1)  X.57390 

The  term  independt^nt  of  -,  is  .3S2G0 

It 

For  this  we  must  substitute  I X  .382G0 

The  equation  will  then  read, 


^=-.98lT  +  J^V  .9C3s|,  +  (.1937  +  (<J-l).5T39)^+(Jx.3826. 
LIMIT   THICKNESS    OF    PIERS. 

43.  ]\[ere  inspection  of  equation  (23)  shows  that  when 
the  height  of  the  pier  becomes  infinite,  we  have 


THICKNESS    OF    PIER. 


231 


tliat  is,  tlie  tliickness  of  the  ])\eY  whose  height  is  infinite 
must  be  equal  to  the  square  root  of  doul)le  the  horizontal 
thrust  niultijilied  by  the  coefficient  of  stability. 

This  interesting  principle  was  discovered  experiment- 
ally by  Eondelet.  It  is  universal — applicable  to  arches  of 
every  form,  aud  under  every  variety  of  circumstances. 
The  moments  of  the  thrust  and  of  the  pier  increase  in 
nearly  the  same  proportion  with  the  height  of  the  pier. 

This  limit  thickness  is  not  very  much  greater  than  the 
thickness  required  for  moderate  heights.  Slightly  chang- 
ing his  radius,  we  take  from  M.  Petit  the  following  illus- 
tration;  r=10';  i?=12'.50;  IC=1.25. 


strict  equilibrium. 

^=2.5000  feet. 

"  2.8190 

"  3.3180 

"  3.6407 

"  3.8657 

"  4.4012 

"  4.9235 

"  5.0687 


Coefficient  of  1.90. 

^=5.6111  feet. 

"  5.7995  " 

"  6.0887  " 

"  6.2640  " 

"  6.3825  " 

"  6.6550  " 

"  6.9147  " 

"  6.9865  " 


7i  =  7.60  feet. 

"     10.00  " 

"     15.00  " 

"     20.00  " 

"     25.00  " 

"     50.00  " 

"  250.00  " 
"  infinite. 

The  whole  increase  of  the  practical  thickness,  ^=1.90, 
from  h=10  feet  to  7^= infinity,  is  but  little  more  than  14 
inches. 

Table  A  gives  the  limit  thickness  for  all  values  of  K 
from  1.10  to  2.00,  both  for  strict  equilibrium  and  for  a 
coefficient  of  1.90—^=1,  and  fcl.90. 

We  thus  have  the  means  of  criticising  many  existing 
cases,  and  may  often  be  spared  much  labor  and  research. 

44.  If  we  wish  to  take  into  account  the  adhesion  of 
mortar,  or  a  load  upon  the  back  of  the  arcli,  the  thrust, 
r%\  which  enters  equations  (22),  (23),  will  no  longer  be 
furnished  by  table  A  alone,  but  must  be  determined  ac- 
cording to  the  circumstances  of  the  case,  as  already  fully 
explained  in  this  section. 
4 


232  THEORY     OF    TUE    ARCH. 

45.  If  we  take  into  consideration  tlie  effect  of  mortar 
only,  the  tlirust  is  given  1  jy  tlie  greatest  of  tlie  two  forces — 

jr^r'C-^rc^^-^;  eq.  (14),  art.  32;  (7  from  column 
4,  table  A. 

r=i-0-rc(K-l)  ;  eq.  (19),  art.  37  ;  O  from  column  5, 
table  A. 

i^  representing  tlie  greatest  of  these  forces,  the  thick- 
ness of  pier  is  given  by  (23),  when  we  have  substituted  ^ 

for  j--C\  or  -2  for  C.     (23)  becomes 


The  effect  of  surcharge  upon  the  thickness  of  pier,  will 
come  up  in  the  discussion  of  other  arches. 


DISCUSSION   OF   TABLE    A. 


4G.  Tliis  table  gives,  either  directly  or  by  proportional 

parts,  for  all  values  of  IC=—  between  1  and  2.732. 

r 

1st.  Tlie  angle  of  inipture. — Rotation. 

2d.  The  ratio,  C\  of  the  thrust  to  the  square  of  the 
radius  of  the  intrados,  in  the  case  of  rotation  and 
the  case  of  sliding. 

3d.  The  ratio,  \/26C,  of  the  limit  tliickness  of  pier  to 
the  radius  of  the  intrados,  for  the  case  of  strict 
equili>)rium,  and  for  the  coefficient  of  stability 
of  l.OO,— fcl,  and  3  =  1.90. 

It  will  be  seen  that  the  angle  of  rupture,  beginning 
with  zero  for  /f=  2.732,  becomes  54°  27'  for  /ir=2.10, 


DISCUSSION     OF    TABLE     A.  233 

attains  its  greatest  value,  64°  9'  for  7r=il.50,  varies  be- 
tween 54°  27'  and  64°  9'  for  all  the  fortification  arches  i  i 
common  use,  that  is,  for  all  the  values  of  K  between  2.10 
and  1.12,  and  ends  with  zero  again  for  K=^\. 

It  will  also  be  seen  that  the  thrust  due  to  sliding  is 
greater  than  the  thrust  due  to  rotation,  for  all  values  of  K 
greater  than  1.44 ;  and  that  the  former  is  less  than  the  lat- 
ter, for  all  smaller  values  of  K.  We  must  in  all  cases,  to 
obtain  the  true  thrust,  select  the  greatest  of  these  two 
values. 

Calling  the  greatest  of  these  values  (or  the  greatest  of 
the  decimals  in  columns  4,  5)  C^  the  limit  thickness  of 
pier  is 

e=?-j/2(7  for  strict  equilibrium. 
^=rj/3.806^for  the  coefficient  of  stabilit}^  1.90. 
e=.r^^hC  for  any  coefficient  of  stability  h. 


As  we  have  ^'/256^=:r/2^X|/6',  it  is  obvious  that, 
while  the  thrust  increases  in  a  geometrical  ratio,  the  limit 
thickness  increases  only  in  an  arithmetical  ratio  ;  and  that 
a  small  error  in  the  thrust  becomes  much  smaller  in  \\\q, 
pier. 

We  are  at  liberty  to  suppose  i\  the  radius  of  the  intra- 
dos,  to  remain  the  same  throughout  the  table.  Assuming 
this,  we  see  that  the  greatest  possible  thrust  that  can  be 
caused  by  rotation  in  any  arch  of  the  radius  r,  is  r^  X  .17535, 
and  corresponds  to  ^^=1.58. 

For  instance,  if  ^=10  feet,  w^e  shall  have  ^^=1.58, 
J?r=15'.80,  and  It — r,  or  the  thickness  of  the  arch  =  5'.80. 

We  must  not  fail,  however,  to  notice  that  the  greater 
thrust,  when  A''=1.58,  is  given  in  the  column  of  sliding. 

47.  By  means  of  this  table,  M.  Petit  has  settled  tliis 
question :  What  is  the  thinnest  arch  that  can  stand  upou 
its  springing  lines  ? 


234 


TDEOUY     OF    THE    ARCH. 


It  is  evidently  necessary  that  the  moment  of  the  thrust 
shoiihl  not  exceed  the  moment  of  the  semi-arch,  l»oth  taken 
in  refei-ence  to  the  exterior  edge  of  the  joint  of  tlie  s])ring- 
ing  lines. 

Now,  the  moment  of  the  semi-arch  is 

The  thrust  is  7-0;  its  moment  i^KC. 


Value  of 

Ratio  of  the  diameter 
to  the  thickness. 

Moment  of  stability  of  the  semi- 
arch  upon  its  springing  lines. 

Monunt  of  the 
horiziintal  tlirust. 

2.000 

2.000 

r^x  2.379075 

7^x0.918240 

1.500 

4.000 

r*  X  0.680956 

7^x0.286250 

1.300 

6.GG0 

r»x  0.305502 

7^x0.186290 

1.200 

lU.OOO 

7^x0.172024 

r»x  0.133608 

1.150 

1.3.333 

r3x0.1l7659 

7^x0.105528 

1.120 

16.60(5 

r*  X  0.088806 

r»  X  0.087763 

1.114 

17.544 

r»  X  0.083320 

r=»  X  0.083434 

1.100 

20.000 

r«x  0.071093 

r^  X  0.074292 

1.050 

40.000 

7-='x  0.031954 

r3  X  0.040034 

1.010 



200.000 

r"  x  0.005844 

7^  X  0.008980 

The  inspection  of  this  table  shows  that  the  arch  has 
aluindant  stability  for  ^=1.30  and  upwards;  that  its 
moment  is  in  e([uilibrium  with  the  moment  of  the  thrust, 
for  7i'=1.114  nearly;  and  that  the  thrust  prevails  more 
and  moi-e  for  smaller  values  of  K.  * 

This  value,  /ir=1.114,  corresponds  to  an  arch  of  which 
the  span  is  a  little  more  than  17A  times  the  thickness ;  and 
that  is  the  thinnest  or  lightest  arch  that  can  possi])ly  stand 
upon  its  .springing  line.  A  thinner  arch  Avould  be  impos- 
sible. 

This  fiict  has  been  confirmed  by  experiment. 


THE    POWDER-MAGAZINE    ARCH.  235 


SECTION  III. 

SEMI-CIRCULAR   ARCHES   OF   180°  WITH  A  ROOF-SHAPED  SURCHARGE 
UPON  THE  CROWN. 

48.  The  powder-magazine  arch,  fig.  9,  belongs  to  this 
class.  To  make  the  inelosure  homb-proof,  a  surcharge  of 
concrete,  or  other  hard  material,  and  sometimes  of  earth, 
also,  is  added  to  the  covering  of  the  arch  proper. 

We  suppose,  for  the  present,  the  plane  of  the  roof  on 
each  side  of  the  central  ridge,  to  be  tangent  to  the  proper 
extrados  of  the  arch,  as  it  frequently  is  in  practice,  and  the 
the  joint  of  rupture  to  rise  vertically  above  the  extrados 
through  the  surcharge.  We  also  suj^pose  the  thrust  to 
act  always  at  a^  the  extrados  at  the  crown,  the  mass  of 
masonry  or  other  material  above  acting  only  as  a  weight. 

Let  Zbe  the  angle  between  the  slope  of  the  roof  and  a 
vertical ;  v  still  the  angle  between  the  joint  of  rupture  and 
a  vertical ;  r  the  radius  of  the  intrados ;  R  the  radius  of 

the  extrados,  supposed    to   be  equally  distant  from  the 

r> 
intrados  throughout ;  K^=i  — . 

r 

We  have,  as  the  general  expression  of  the  horizontal 
force  required  to  keep  any  voussoir  A  ah  m  np  A'  from 
falling  by  rotation  round  the  point  m, 

+  ,))-(J^— t-^)r  (24) 

^/      \sm.  t»     cos.^'fiy  j 

*  Fia;  9.     CA'  =  -. — ^  =  B ;  nn=. ^ — '— =B' ;  Rsin.v=h  =  perpendic- 

sin.  /  sin.  /  '     ' 

ular  distance  between  GA'  and  np;  h—-=, — -— =  tlie  distance  of  the  center  of  jrravitv 

^      Z{B-{-B)  " 

of  CA' p  n  from  C A' ;  h  x f  »-8in.  y  — A^-— -  -^  Wmoment  of  trapezoid  CA' j)  ii 

on  »?=1   , J-         '  X  E  sin.2  vxdr-R^  sin."  vi : J 1=-^  r'ein.^y 

*  \  sin.  /  ^  sin.  /  //     ° 


236  THEORY    OF    THE    ARCH. 

Tlie  maximuni  value  of  tins  expression  must  be  found 
for  any  i)ai-tieular  arcli  by  trial. 

M.  Petit  makes  an  aj)plication  of  this  formula,  to  tlie 
powder  magazine  of  Vauban,  of  the  following  dimensions, 
viz.:  7=49°.  7'.  17";  i?=5.035  metres=16.5148  feet; 
7—4.0605  metres=13.3184  feet;  Ji  =  1.2'i. 

Tliese  values  of  /  and  JT  substituted  in  the  preceding 
equation,  give 

for  v^5^\  .  F' =0.229200 Xr^; 
"  ?'=54\.i^'=0.229381X/'=i^; 
"  v=5'o\.  F' =0.229295 Xr\ 

Near  the  angle  of  maximum  thrust,  and  as  far  as  six  or 
eight  degrees  on  both  sides  of  that  angle,  the  variations  of 
F'  are  very  small.  The  exact  determination  of  that  angle 
is  not  important.  It  may  be  regarded  in  the  present  case 
as  54°.     The  thrust  is  ^"1=0.229381  X :^-^=:40.69  feet. 

If  one  cubic  foot  of  the  masonry  Aveigh  150  pounds,  the 
thrust  is  equivalent  to  the  effort  necessary  to  sustain  40.G9 
Xl50  p(ninds=(n03  pounds. 


EFFECT   OF    :MOPvTAE    UPON    THE    KOTATION   THRUST. 

49.  To  obtain  the  thrust  diminished  by  the  adhesion 
of  mortar  upon  the  joints,  we  have  only  to  suljtract  from 
the  value  of  the  second  member  of  (24),  corres2)onding  to 
each  .assigned  value  of  v,  the  expression  (see  equation  (3) 
and  art.  15,  31), — 

J (6  — SA'— (3  — 'JA')8in.  I+v)  I  .     Moment  of  sector,  C'iwi,  on  7n=iivrHr  b\d.  v 

(  sin.  /  j  V 

r'J  pin.' ir\      .  ,    .    „    /    8»;            1      \  .  pin.'w 

-|x ^-l^Jz-'sin."!.^, ^~-];  4  8in.M''  =  —  ,r;F'xatn'=rxr 

(A'— COS.  t;)=momcut  of  trapezoid  C  A'  p  7i— moment  of  sectoi*C'6  m  ;  hence  F'=.&i 
above,  (24). 


EFFECT    OF    MORTAR.  237 


Qj/  3(/ir— COS.  v) 

The  resulting  maximum  will  be  tlie  true  tlirust. 

But  we  can  simplify  tliis  expression,  as  in  art.  32,  by 
supposing  V,  in  tlie  term  involving  the  adhesion  of  mortar, 
to  be  always  60°,  which  differs  but  little  from  the  angle  of 
maximum  thrust  in  heavy  semi-circular  arches,  however 
loaded.     See  the  various  tables  contained  in  this  paper. 

Suppose  we  have  obtained  by  direct  calculation,  or 
from  tables  A,  C,  D,  or  F,  the  thrust  ?'^  (7  without  regard 
to  mortar.  The  thrust  diminished  by  mortar  is  given  by 
equation  (14), 

F=r' 0-irc^^i~^J  (14) 

3      2/1^-1  ^     ^ 

The  effect  of  adhesion  depends  solely  upon  the  dimen- 
sions of  the  arch  proper,  and  is  not  affected  by  its  load.  It 
is,  therefore,  the  same  in  all  equal  arches  however  different 
may  be  their  loads  ;  and  the  general  conclusions  of  art.  33 
are  applicable  to  all  arches,  as  already  stated. 

If  the  joints  are  unequal,  as  often  hap]3ens,  we  may 
still  suppose  'y=60°,  and  d'  representing  the  length  of  the 
joint  of  rupture,  d  the  length  of  the  vertical  joint,  the 
thrust  will  be 

F=r^C-'^^f+p^  (14)' 

O  being  the  number  which,  multiplied  by  ?'^,  gives  the 
thrust  independently  of  mortar ;  c  the  force,  in  pounds, 
required  to  separate  a  square  unit  of  the  joint,  divided  l>y 
the  weight  of  a  cubic  unit  of  masonry. 

Example :  the  magazine  of  Va'uban,  art.  48. 

j^^r'^C-h'c^^^~^^' =?■'  X  0.229381  -r  X  0.519=  40.69 
^      2/i"-l 

feet  —  6.91  feet  =  6103  pounds  —  1036  pounds  =  5067 
pounds. 


238  THEORY  OF  THE  ARCH. 

TUE  EFFECT  OF  SURCHARGE  UPON  THE  ROTATION  THRUST. 

50.  To  ol)tain  the  thrust  a.s  modified  l)y  a  load  or  sur- 
charge of  the  density  of  the  iiia^oiiry,  and  of  tlie  constant 
vertical  depth  /,  we  must  add  to  the  second  member  of 
(24),  see  art.  34, 

S'p'_rtIC(2-X)        sin.'^  v 
y  2  A''— COS.  v 

of  which  the  maximum  value  is,  art.  34, 

The  effect  of  this  surcharge  depends  solely  upon  the 
dimensions  of  the  arch  proj^er,  and  is  not  affected  by  any 
other  load.  Suppose  we  have  obtained  liy  calculation,  or 
from  tables  A,  C,  D,  or  F,  the  thrust  r'^0  without  regard 
to  surcharge,  we  have 

F^r-C^rtN.  (15)' 

We  have  given,  in  the  13th  column  of  table  F,  the 
values  of  iV  corresponding  to  all  values  of  A"  between  1.02 
and  1.42,  applicable  to  all  semi-circular  arches,  whatever 
load  they  nuiy  carry  in  addition  to  the  surcharge  of  con- 
stant depth. 

This  mode  of  treating  the  surcharge,  leads  to  a  small 
error  in  excess,  or  in  favor  of  sta])ility;  for  while  r^Ci^  the 
thrust  indej)endently  of  surchai'ge,  calculated  at  a  particu- 
lar angle,  rtN  is  the  maximum  effect  of  the  surcharge,  cor- 
responding, generally,  to  some  other  angle. 

The  true  thrust  would  correspond  to  some  intermediate 
angle,  and  would  ])e  somewhat  less  than  the  sum  of  the 
two  maxima.  Tlic  difference,  however,  in  tlie  two  angles, 
is  very  small ;  in  the  heavy  arches  of  fortifications  never 
exceeding  six  or  eight  degrees. — See  taljle  F.  The  error, 
therefore,   always  in  excess,  is  very  small,  it  being  the 


THE    MAGAZINE    ARCH.  239 

property  of  a  maximum  to  exceed  but  little  the  neigliT)oi'ing 
values  of  the  same  function.  This  princi])le  has  been  illus- 
trated in  art.  29. 

51.  The  weight  of  a  single  column  resting  upon  the 
crown  of  the  arch,  we  can  always  represent  by  a  surface 
one  unit  in  width  and  II  m  height. 

The  thrust  increased  by  such  a  weight,  see  art.  35,  is, 

TT 

_F=7-'^CA-         ,  r'^C  beins:   the   thrust   independently 

of  surcharge,  obtained  from  table  F  or  by  direct  calcul- 
ation. 

For  a  more  extended  discussion  of  this  case,  see  art.  35. 
All  the  remarks  there  made  are  equally  applicable  here. 


THE   SLIDING   TIIKUST. 

52.  The  general  expression  of  the  horizontal  force 
necessary  to  prevent  the  segment  whose  surface  is  8^  from 
sliding  down  its  lower  joint  n  w,  is,  equation  (8),  P'=S 
cot.  {a-\-v)  ;  which,  in  the  present  case,  becomes,  fig.  9, 

F'=r'  .       ,  '"''-^     ,     ,  i  A-Yl-^sin.(/+.))-isin./-A-  I*  -(25) 
sin. /tang,  (a +  1^)  (        V       -         ^         v      -  &m.v  \ 

Assuming  «=30°,  /  and  ^  being  known  by  the  con- 
ditions of  the  problem,  we  must  find,  by  assigning,  in  suc- 
cession, different  values  to  v,  the  maximum  value  of  P'  in 
(25). 

The  greatest  of  the  two  maxima,  F,  P,  (24),  (25),  wdl 
be  the  true  thrust. 


^      /    R       E—R  sin.  (r+v)\     „  . 
\sm.  /  sill.  J  / 


240  THEORY     OF    THE    ARCH. 


TIIK    KFFECT    OF   MORTAR   UPON   TDE    SLIDING    THRUST. 

53.  The  slulini?  tlini^it  (lliniiiislied  by  the  adhesion  of 
mortar,  is  the  maximum  value  of  P'  in  equation   (9) — 

7^'  =  .S'eot.  (a-\rv^ — ^— r — — t"-     The  ancfle  t' which  ren- 
^   ^  ^     sm.  {a-\-v)  ^ 

ders  S  cot.  {a-\-v)  a  maximum,  or  whicli  corresponds  to 
the  true  sliding  thrust  independently  of  mortar,  varies  in 
the  several  arches,  from  29°  when  the  roof  is  horizontal,  to 
22°  Avhen  tlie  roof  is  inclined  45°  on  each  side  of  the  cen- 
tral ridge.  As  in  art.  37,  we  can,  without  sensible  error, 
su})]i(\se  i\  in  the  tei-m  involving  the  adhesion  of  mortar, 
always  equal  to  30°.  We  have  then,  this  formula  for  the 
sliding  thrust  diminished  by  the  adhesion  of  mortar, 

in  which  C  is  taken  from  table  C,  D,  or  F,  or  obtained  by 
direct  calculation,  and  tlie  last  term  is  the  effect  of  mortar. 
For  illustration  of  this  subject,  see  art.  37. 

The  effect  of  adhesion  depends  solely  upon  the  length 
of  the  joint,  and  is  therefore  the  same  in  all  equal  arches 
however  loaded. 

The  above  formula  Avill  seldom  l)e  needed ;  for  it  is 
only  in  very  heavy  arches  that  the  sliding  can  exceed  the 
rotation  thrust,  and  in  such  arches  it  would  hardly  be  safe 
to  rely  upon  any  adhesion  of  mortar. 


EFFECT   OF   SURCHARGE    UPON   THE   SLIDING   THRUST. 

54.  The  general  equation  applicable  to  this  case  is 
F^H  C(jt.  (y-f-30°)+/S"  cot.  (t'+30°).  We  can  always 
suppose  tlie  surcharge  ^S^',  wliich  is  entirely  above  the  roof 
of  the  arch,  to  have  a  constant  vertical  depth  as  fjir  as  25° 
or  30°  from  tlie  crown.     Let  t  be  that  depth.     We  have 


THE    MAGAZINE    ARCH.  241 

jS'=txIix sin.  V = rtir sin.  v  ;  and  ;S"cot.  {v + 30°) = vtK 
sin. '^Xcot.  ('y+30°),  of  wliicli  the  maximum  value,  corre- 
sponding very  nearly  to  t;=25°,  is,  rtKx  0.29592.  Adding 
the  two  maxima  together,  we  have,  with  a  slight  error  in 
favor  of  stability, 

P=r2(9'_j_,,^7i-X  0.29592  (20)' 

r^ (7  being  the  sliding  thrust  independently  of  surcharge, 
obtained  from  tables  C,  D,  or  F,  or  by  direct  calculation. 
This  formula  has  already  been  given,  art.  89,  equation  (20), 
and  is  applicable  to  all  circular  arches  however  loaded — 
always  giving  a  result  slightly  in  excess. 

We  have  given  in  table  F  the  value  of  ^x  0.29592, 
corresponding  to  values  of  K  which  render  the  diding 
greater  than  the  rotation  thrust^  applicable  to  all  semi- 
circular arches,  tables  A,  C,  D,  F,  and  to  segmental  arches 
which  have  the  angle  at  the  center  greater  than  25°. 

This  mode  of  treating  the  surcharge  is  precisely  like 
that  adopted  in  art.  50.  Here  too,  the  error,  always  in 
excess,  is  very  small;  for  the  angle  t',  which  gives  the 
sliding  thrust  without  regard  to  surcharge,  varies,  in  the 
several  arches,  from  22°  to  29°,  never  differing  more  than 
4°  from  the  angle  25°,  which  renders  the  effect  of  the  sur- 
charge a  maximum. 

The  true  thrust  will  be  the  greatest  of  the  two  maxima 
F,  F,  (15)',  (20)'. 


GENERAL  REMARKS  ON  TABLES  C,  D,  E. 

55.  M.  Petit  has  applied  the  general  equations  (24), 
(25),  to  the  two  extreme  cases  in  which  the  roofs  are, 
respectively,  horizontal  and  inclined  45° ;  limits  between 
which  probably  all  powder-magazine  arches  are  comprised. 
Tables  C  and  D  contain  his  results.  We  have  filled  up 
the   interval,   between   7=45°    and   /=90°,   with    eight 


242  TUEORY    OF    THE    ARCH. 

column"*,  /  varyini;  at  intervals  of  5°.  Tlie  results  are 
embodied  in  talile  F,  which  gives  directly,  or  by  i)ropor- 
tional  part-5,  the  thrust  of  all  circular  arches  of  common 
use,  whose  loads  are  bounded  on  top  by  two  symmetrical 
planes  extending  as  for,  at  least,  as  from  the  crown  to  the 
reins,  or  angle  of  rupture — applicable  to  all  semi-circular 
roof-covered  magazine  arches,  and  to  almost  all  full-circle 
stone  and  brick  bridges.     See  discussion  of  that  table. 

For  the  purpose  of  general  illustration,  and  as  intro- 
ductory to  tables  C  and  D,  we  give  the  application  of 
(24),  (25),  to  the  two  cases  mentioned  above,  /=45°,  and 


EOOF   INCLINED    FORTY-FIVE   DEGREES. 

56.  Make  /=45°  in  equation  (24).     It  becomes.. 

—  Sin    V         (  — 

^='-'0{A-r;S:iT    ^V5(0-3A--(3-2^)  sio.(45°  +  .))- 

,Jv_ 1_\  >  (2«) 

\sm.v     cos.'^  v/  j 

Suppose,  furthermore  a=SO°  in  equation  (25),  and  for 
V  substitute  22°,  which  corresponds,  very  nearly,  to  the 
maximum  value  of  P',  or  the  horizontal  sliding  thrust,  we 
have 

P=r-(0.2234x/r'-0.14999)  (27) 

By  means  of  these  two  fornmlie,  and  of  another, 
€=7'^2bCj  art.  43,  M.  Petit  has  calculated  table  C,  ana- 
logous to  table  A,  giving,  for  all  values  of  IC  between 
1.05  and  2,  the  angle  of  rupture,  the  thrust — rotation  and 
slidinjir,  and  the  limit  thickness  of  pier  recpiired  for  strict 
equililjrium  and  for  the  co-efficient  of  stability  2.  See 
discussion  of  tliat  table. 

M.  Petit  has  ])roved  conclusively,  by  a  coui'se  of 
reasoning  like  that  given  in  art.  47,  that  this  kind  of  arch, 


SURCHARGE     IIORIZOKTAL.  243 

7=45°,  is  always  stable  upon  its  springing  line;  that  is, 
that  however  thin  the  arch  may  be,  the  moment  of  the 
thrust  is  always  less  than  the  moment  of  the  semi-arch  in 
relation  to  the  exterior  edge  of  the  springing  line. 


EOOF   HORIZONTAL. 

Figure  4. 
57.  Make  7=^90°  in  equation  (24).     It  becomes 

Make  7=90°,  «  =  30°,  in  (25),  and  for  v  substitute  29°, 
which  corresponds  very  nearly  to  the  maximum  value  of 
P\  we  have 

P=?X0.16391x7r2_o;^52Qg>j  ^29) 

By  means  of  these  two  formulae,  and  of  another — 
6=r|/2^6^,  M.  Petit  has  calculated  table  D  analogous  to 
tables  A  and  C.     See  discussion  of  that  ta1  )le. 

By  a  table  similar  to  that  given  in  47,  M.  Petit  has 
demonstrated  that  the  moment  of  the  semi-arch  exceeds 
the  moment  of  the  thrust,  both  taken  in  reference  to  the 
exterior  edge  of  the  springing  line,  for  all  values  of  K 
greater  than  1.0435;  and  that  the  moment  of  the  thrust  is 
the  greater  for  all  smaller  values  of  K.  The  thinnest  arch 
therefore,  of  this  kind,  having  a  surcharge  limited  to  the 
horizontal  line  tangent  to  the  extrados  at  the  crown,  Jfigs. 
4,  7,  that  can  stand  upon  its  springing  lines,  is  one  whose 
span  is  about  46  times  its  thickness. 


0  44  THEORY     OF    THE     ARCH. 

DISCUSSION    OF   TABLE    C THE    KOOF   INCLINED    FORTY-FIVE 

DEGREES. 

58.  After  explaining  tlie  use  of  tables  C,  D,  and  F,  we 
shall  show  liow  to  apply  tlie  results  to  the  determination 
of  the  thickness  of  pier. 

Tal)le  C  gives,  either  directly,  or  hy  proportional  parts, 
for  all  values  of /i^  between  1.05  and  2  : — 

1st.  The  angle  of  rupture,  or  of  maximum  thrust. — 
Rotation. 

2d.  The  decimal  (7,  column  4,  which,  multiplied  by  the 
square  of  the  radius  of  the  intrados,  gives  the  horizontal 
thrust  on  the  supposition  of  rupture  by  rotation . 

3d.  The  decimal  6',  column  5,  which,  multiplied  by 
r^,  gives  the  thrust  on  the  supposition  of  rupture  by 
sliding. 

4th.  Columns  6,  Y ;  the  value  of  the  radical,  \/  26C\ 
which,  multiplied  by  the  radius  of  the  intrados,  gives  the 
limit  thickness  of  pier  in  the  case  of  strict  equilibrium, 
fcl,  and  with  the  "stability  of  Vauban,"  ^  =  2. 

It  will  be  seen  that  the  thrust  due  to  sliding  is  greater 
than  the  thrust  due  to  rotation  for  all  values  of  Jl  greater 
than  1.42,  and  that  the  rotation  thrust  is  the  greater  for 
all  smaller  values  of  IT. 

Calling  the  greatest  of  these  values  (or  the  greatest  of 
the  decimals  in  columns  4,  5)  0  the  limit  thickness  of  pier, 
or  the  thickness  required  for  an  infinite  height,  is 

e=rf/26';  for  strict  equilibrium  ;  S=l. 
e=ir\^4:0,'  for  the  stability  of  Vauban  ;  5  =  2. 

Tlie  greatest  of  the  decimals  in  columns  4,  5,  must 
always  be  selected  as  giving  the  true  thrust. 

The  thrust  given  in  table  C  for  ^=1.20,  is  double  the 
thrust  in  ta>)le  A  for  the  same  value  of  II.  For  larger 
values  of /iTthe  thrusts  of  table  C  are  more  than  double, 


TABLES     C.     AND    D.  245 

and  for  smaller  values  of  K  less  tlian  douLle,  the  tlirusts 
of  table  A. 

Of  all  arches  having  the  same  span  and  radii,  the 
magazine  arch  has  the  greatest  thrust.  Its  span,  however, 
is  small,  seldom  exceeding  30  feet. 


DISCUSSION    OF   TABLE   D THE    SUECHARGE   HORIZONTAL. 

59.  Table  D,  analogous  to  A  and  C,  gives  directly  or 
by  proportional  parts,  for  all  values  of  K  between  1 
and  2. 

1st.  The  angle  of  rupture,  or  of  maximum  thrust. — 
Rotation. 

2d.  The  decimal  C,  columns  4  and  5,  which,  multiplied 
by  the  square  of  the  radius  of  the  intrados,  gives  the 
horizontal  thrust  on  the  supposition  of  rupture  by  rota- 
tion, and  by  sliding. 

3d.  The  value  of  the  radical  |/25(7,  columns  6,  7, 
which,  multiplied  by  the  radius  of  the  intrados,  gives  the 
limit  thickness  of  pier  in  the  case  of  strict  equilibrium, 
l—\  and,  with  the  "stability  of  Lahire,"  ^=^1.90. 

It  will  be  seen  that  the  thrust  due  to  sliding  is  greater 
than  the  thrust  due  to  rotation  for  all  values  of  K  above 
1.34,  and  that  the  former  is  less  than  the  latter  for  all 
smaller  values  of  K.  We  must  in  all  cases,  to  obtain  the 
true  thrust,  select  the  larger  of  these  tw^o  values. 

The  angle  of  rupture  for  all  the  usual  values  of  A"  in 
which  the  thrust  is  due  to  rotation,  that  is,  from  A^=1.10 
to  A"=1.34,  differs  from  60°  by  only  5°;  while,  for  the 
most  common  values  of  K^  the  diiference  is  less. 

The  thrust  given  by  table  D  is  equal  to  the  thrust 
of  table  A  when  ATr^  1.30. 

The  thrusts  of  table  D  exceed  those  of  table  A  when 
AT  is  less  than  1.30. 


240  THEORY     OF    THE    ARCE. 

The  thrusts  of  table  A  exceed  those  of  table  D  when 
K  is  greater  than  1.30. 

The  greatest  possible  thrust  that  can  be  caased  by 
rotation  in  any  arch  of  the  radius  i\  is 

^•■X0.1450G,  and  corresponds  to  A"=1.36. 

The  thrust  of  talJe  C  is  about  double  the  thrust  of 
table  D  when  7r=1.34. 


DISCUSSION   AND    USE   OF   TABLE   F. 

60.  Columns  3  and  12  have  been  extracted  from  tables 
C  aud  D,  calculated  by  M.  Petit.  The  remaining  columns 
have  been  calculated  for  this  paper. 

Tliis  table  gives  directly,  or  by  proportional  parts,  the 
thrust,  for  all  values  of  JT  between  1.02  and  1.50,  of  all 
semi-circular  arches  which  cany  loads  of  masonry,  or  of 
ecpial  weight  with  masonry,  rising  to  two  planes  meeting 
along  a  central  ridge,  and  tangent  to  the  extrados  of  the 
arch,  or  to  surfaces  parallel  to  such  tangent  planes. 

Column  3  gives  the  thrust  for  that  case  in  which  the 
two  planes  become  one  and  horizontal,  tangent  to  the 
extrados  at  the  crown. 

Column  4  gives  the  thrust  for  /=85°,  the  two  planes 
making  each  an  angle  of  5°  with  a  horizontal,  and  being 
each  tangent  to  the  extrados  5°  from  the  vertical  or  central 
joint. 

Column  5  gives  the  thrust  for  /=80°,  the  two  planes 
making  each  an  angle  of  10°  with  a  horizontal  and  touch- 
ing the  extrados  10°  from  the  summit. 

Columns  6,  7,  8,  0,  10,  11,  and  12,  give  the  thrust  for 
/=successively,  75°,  70°,  65°,  60°,  55°,  50°,  and  45°. 

Column  13  gives  the  addition  to  the  thrust  (rotation) 
caused  Ijy  a  surchai-ge  of  uniform  vertical  depth  t  above 
the  roof. 


EXPLANATION     OF    TABLE    F.  247 

Colnmn  14  gives  in  like  manner  the  addition  to  t]ie 
sliding  tlinist  caused  by  a  surcharge  of  uniform  deptli,  ^, 
above  the  roof. 

Near  the  top  of  each  cobimn  of  thrusts,  will  be  seen  a 
horizontal  line.  Below  that  line,  the  true  thrust  is  due  to 
rotation ;  above,  to  sliding.     When  tlie  ratio  of  the  two 

radii,  —=.K^  places  the  thrust  below  that  line,  the  addi- 
tion A,  caused  by  surcharge,  must  be  taken  from  column 
13.  If  the  thrust  is  found  above  the  horizontal  line,  A 
must  be  taken  from  column  14. 

Columns  3,  9,  and  12  give  the  angles  of  maximum 
thrust :  for  sliding,  down  to  the  horizontal  line  ;  for  rota- 
tion, below.  These  angles,  for  want  of  room,  are  not  given 
in  the  other  columns. 

In  columns  3,  9,  and  12,  the  thrusts  are  calculated 
within  \  degree  of  the  angle  of  maximum  thrust ;  in  the 
other  columns  within  one  degree. 

Columns  13  and  14  give  results  generally  a  little  in 
excess,  never  too  small. 

Column  13  gives  the  angles  corresponding  to  the  maxi- 
mum effect  of  the  surcharge  in  the  case  of  rotation. 
Comparing  these  angles  with  the  angles  of  maximum 
thrust,  in  columns  9  and.  12,  we  see  that,  from  A^=1.10  to 
-Zr=  1.42,  they  differ  in  no  case  more  than  9°,  while  the 
mean  difference  is  much  less.  We  therefore  know  that  the 
error  in  excess,  arising  from  adding  these  two  maxima 
together,  is  exceedingly  small.  The  difference  in  these 
angles  is  very  small  in  all  the  magazine  arches  in  common 
use,  never,  it  is  believed,  exceeding  7°.  It  is  only  in  very 
light  arches,  A^i=1.02,  1.03,  &c.,  loaded  nearly  or  quite 
horizontally,  that  the  error  becomes  of  any  consequence. 

The  error  in  excess  in  using  column  14,  is  never  of  any 
consequence, 

5 


24S  THEORY    OF    THE    ARCH. 

lU'LES    roil    USIXG    TAliLE    F. 

CI.  From  tlie  dimensions  of  tlie  given  arcli,  determine 

It 
the  ratio  —  ^A",  tlie  angle  I^  and  the  elevation  E  of  the 

ridge  above  the  springing  line.  If  A'  and  /  or  E^  are 
found  in  the  table,  the  thrust  is  given  at  once  at  the  inter- 
section of  the  two  columns.  But  one  or  both  of  these 
quantities,  K^  and  /  or  E^  Avill  generally  be  intermediate 
between  the  tabnh\r  numbers.  The  thrust  must  then  be 
determined  by  proportional  parts. 

AVe  sup])ose  the  thrust  to  vary  uniformly  between  any 
two  successive  horizontal  columns,  say  from  /r=:1.20  to 
K^=i\jl\  ;  and  also  uniformly  between  any  two  successive 
vertical  columns,  say  from  /=r50°  to  jr=:45°,  or  rather 
from  A'=i?X  1.30541  to  j^r=7?X  1.41421.  This  sort  of 
interpolation  can  offer  no  difficulty  to  those  who  under- 
stand common  arithmetic. 

Rule  I.  Suppose  7J  of  the  given  arch,  to  have  one  of 
its  values  in  table  F,  K  being  between  two  tabular  values 
of  that  ratio. 

Under  the  given  value  of  /,  take  the  difference  of  the 
decimals  opposite  the  adjacent  values  of  K.  Call  that  dif- 
ference dd  (ditlerence  of  decimals);  let  <:Z /iT represent  the 
difference  between  the  adjacent  values  of  K^  and  d'  K\\\^ 
excess  of  the  given  value  of  A' over  the  adjacent  smaller 
value  of  K.     The  proportion 

dK:dd'.'.d'  K'.x 
gl\es  the  correction;  which  nmst  be  added  to  the  decimal 
under  /  and  op])osite  the  smaller  adjacent  value  of  7^, 
■when  the  decimals  increase  ascending ;  subtracted  when 
they  decrease  ascending. 

Rule  II.  Suppose  -ST,  of  the  given  arch,  to  have  one  of 
its  values  in  tal)le  F,  /  or  A' coming  between  two  tabular 
values  of  those  quantities. 


USE     OF    TABLE    F.  249 

Opposite  the  given  value  of  K^  take  tlie  difference  of 
tlie  decimals  under  tlie  greater  and  less  adjacent  values  of 
I  ov  K  Call  tliat  difference  c/ (/.  Let  f^/i' represent  the 
difference  between  the  adjacent  values  of  E^  and  d!  ^the 
excess  of  the  given  value  of  E  over  the  adjacent  smaller 
value.     The  proportion 

d  E:  d  d  : :  d'  E:  x 
gives  the  required  correction;  which  must  l)e  added  to 
the  decimal  opposite  K  and  under  the  smaller  adjacent 
value  of  E^  when  the  decimals  increase  from  left  to  right ; 
subtracted,  when  they  decrease  from  left  to  right.  We 
may  use  I  or  E  according  to  convenience.  Using  the 
former,  the  proportion  will  be 

5°  :  dd  '.'.  d'  I'.  X. 

The  results  will  be  nearly  the  same.  As  I  diminishes 
from  left  to  right,  dH^  in  this  last  proportion,  will  be 
the  excess  of  the  preceding  adjacent  value  of  I  over  the 
given  value  of  7,  exj)ressed  in  degrees  and  decimals  of  a 
degree. 

Rule  III.  Suppose  E  and  I  both  to  be  intermediate 
between  tabular  values  of  those  quantities. 

By  rule  I.  determine  the  decimal  corresponding  to  the 
given  value  of  .ZTand  the  preceding  adjacent  value  of  Zor 
E j  then  by  the  same  rule  determine  the  decimal  corre- 
sponding to  the  given  value  of  K^  and  the  following  ad- 
jacent value  of /or  E.  Call  the  difference  between  these 
two  decimals  dd.  Let  dE  represent  the  difference  be- 
tween the  adjacent  values  of /^,  and  d'  E\\\q  excess  of  the 
given  value  of  E  over  the  preceding  adjacent  smaller 
value.     The  proportion. 

dE:dd'.:d'E'.x, 

gives  the  required  correction,  to  be  added  when  tlie 
decimals  increase  from  left  to  right,  subtracted  when  they 
decrease  from  left  to  ri2:ht. 


250  TnEORY     OF    THE    ARCH. 

If  we  iis^e  I  instead  of  E,  tlie  proportion  will  be 
h°  '.ddWil  I\x. 

Calling  C  the  final  value  of  tlie  decimal,  we  have  the 
thrnstP=;-x6'. 

In  i)raetice,  the  roof  is  hardly  ever  more  inclined  than 
45°.  Still,  we  can,  if  necessary,  obtain  from  the  talde  the 
thrust  eorrespondino:  to  smaller  values  of  /,  on  the  prin- 
ciples above  explained. 

The  rate  of  variation  from  7=50°  to  7=45°,  we  may, 
without  sensible  error,  suppose  continued  a  few  degrees 
below  45°. 

Kule  W .  If  there  l)c  a  surcharge  in  masonry  rising  to 
the  lines  1{  I)\  FJ  F\  fig.  10,  draw  the  parallel  lines  RD, 
R  TJ  tangent  to  the  extrados  of  the  arch. 

Then  determine  the  thrust  of  the  arch  It  h  A  E  D  R^ 
limited  l>y  these  tangent  lines,  by  the  rules  already  given ; 
and  add,  for  surcharge,  from  columns  13, 14,  as  directed  in 
tl^ose  columns.  The  addition  to  be  made  is  A  —  rtX 
decimal,  in  which  t—R'R—D'D\  7'=tlie  radius  of  the 
inti-ados.  The  decimals  in  columns  13,  14,  are  the  same 
for  all  values  of  7,  but  are  supposed  to  increa.se  or 
diminish  uniformly  between  any  two  consecutive  values  of 
K.  They  must  be  corrected  l)y  rule  I.  when  the  ratio,  K^ 
of  the  ofiven  arch  Is  not  found  in  the  table. 

Rule  V.  If  there  be  a  surcharge  of  earth  aT)Ove  the 
masonry,  reduce  its  thickness  over  the  key  and  over  the 
reins,  in  the  proportion  of  its  density  divided  by  the 
density  of  the  masonry. 

We  can  then  regard  all  that  comes  below  the  reduced 
line  as  (•omi)osed  of  niasonry  alone,  and  the  case  falls  under 
rule  IV. 

62.  If  the  two  sides  of  the  arch  be  unequally  loaded, 
look  for  the  thrust  on  that  side  wliich  cariies  the  greatest 
load, — for  the  thickness  of  pier  on  that  side  which  carries 
the  least. 


USE    OF    TABLE    F. 


251 


63.  Example  1.  The  magazine  of  Vauban. 

^— 13'.3184;    ^  =  1G'.5U8;    lieuce   A'=:-  =  1.24   very 

nearly;  7=49°  7' V"  ;  j5'=21'.85  =  i?X  1.32258. 

Referring  to  rule  II.  and  to  columns  ^=1.24  ;  7=45°, 
7=50°,  we  liave 

(IE  dd  d' E 

1.41421         .26850         1.32258 
1.30541         .22219         1.30541 


.10880 


.04631  :  :      .01717 


a'r=.00731 
.22219 


•     ^  F^rr^x  0.22950 
M.  Petit  gives,  from  an  independent  cal- 
culation,         F=y-X  0.22938 

Tlie  difference  ^'2X0.00012 
is  in  effect  nothing.  Had  Ave  interpolated  in  reference  to 
7,  the  result  would  not  have  been  quite  so  accurate,  as  the 
thrust  varies  rather  with  the  elevation  of  the  ridge  than 
with  the  angle  I. 

Example  2.  The  magazine  at  Fort  Jefferson. 

'/'=14';    i?=:17'.50;    7i"=il.25  ;    tang.  7=^- ;  7r=56°.  3' 

'         ^  17.5' 

23";  7'=^:^^=^Xl.20542  =  21'.0948. 
sm.  7 

Referring  again  to  rule  11.  and  to  columns  7i"=:1.25, 

7=60°,  7=55°,  we  have 

dE  dd  d'E 

1.22078         .19027  1.20542 

1.15470         .16492  1.15470 


.06608 


.02535 


.05072  :  a?=0.01946 
0.16492 


The  thrust  is  .  .  ^-2x0.18438=36.13848 


'2o'2  THEORY     OF    THE    ARCH. 


Bret,  foi-ward,  36.13848 
To  tills  we  must  add  for  a  surcharge  5'.90  deep 
(see  culuuni  13,  opposite  /r=:1.25), 

.1  =  14.  X  5.90  X0.46875  =  38.'n875 


Giving  as  the  total  thrust,  say,  .  F= 74.86000 

Exaini)le  3.  An  arch  constructed  at  Fort  Porter,  fig.  11. 
This  arch  has  a  surcharge,  both  of  masonry  and  earth. 

Reducing  the  height  uf  the  latter  in  the  proportion  of 
2  to  3,  supposed  to  be  the  relative  weights  of  a  cubic  foot 
of  earth  and  masonry,  we  find  the  reduced  ridge  to  be 
14'. 2 8  above  the  springing  line,  and  the  reduced  roof,  on 
the  side  most  loaded,  to  make  an  angle  of  86°  46'  with  a 
vertical  line.     The  data  are 

/•=6';    i?=7'.66S;    7^=1.278;  7=86°  46';    E=-^-r^ 
'  '  '  '  sm.i 

7?Xl.00137=:7'.68  ;  ?'=14'.28-^=6'.60. 

This  case  comes  under  rules  III.  IV.  V. 
By  rule  I. 

dK=l :  77=.00085  : :  d'K=.^  :    a'=0.00068 

Add  .  .  0.14101 


Giving,  as  the  decimal  for  7^=:  1.278, 

7=90°, 0.14169=0.14169 

Find,  in  like  manner,  the  decimal  for 

7i."=1.278,  7=85°,         .         .         .     0.13486 


77=  0.00683 
77i''=.00382  :  77=.00683  : :  77i'=.00137  :  x=  .     0.00245 


Giving  as  the  thrust,  without  surcharge,    ^■'^X  0.13924 

=  5.01204 
To  this  add  for  surcharo^e  ^4  =  6  X  6.60  X  0.44495  =  17.62000 


Total  thrust,     =F=22.63000 


THICKNESS    OF     PIER.— GENERAL    FORMULAE.  253 

THICKNESS     OF    PIER,    THE   ROOF    HAVING    ANY    INCLINATION. 

Figure  9. 
G4.  We  suppose  tlie  surcharge,  if  ])artly  of  eartli  or 

any  light  matenal,  to  be  reduced  in  height  in  the  propor- 
tion of  its  density  compared  with  the  density  of  the  arch 

proper.     Let 

A=:the  mean  height  of  the  pier  from  its  base  to  the  sur- 
face of  the  reduced  surcharge  over  its  top  =  j^'7>,  fig. 
4  ;   0  0\  fig,  10  ;  to  be  estimated  if  not  known, 

jB'=the  elevation  of  the  reduced  ridge  above  the  springing- 
line=(7^,  fig,  4;   C E\  fig.  10,  11.' 

^'=the  depth  of  the  arch  and  its  reduced  surcharge  at 
the  springing  line =^4  a  fig.  10  ;  =^Ca  or  JE^  fig.  4. 

w=the  surface  of  that  part  of  the  arch  and  its  reduced 
load  which  lies  directly  over  the  half-span.' 

mz=zt\\Q  moment  of  that  surface  in  relation  to  the  interior 
edge  of  the  joint  of  the  springing  line. 

^=the  lever-arm  of  the  thrust=«  ^,  figs.  4,  10. 

J^^=the  horizontal  thrust  however  determined. 

7'=:the  half  span = the  radius  of  the  intrados  in  semi- 
circular arches. 

a?=:the  distance  between  the  exterior  edge  of  the  base 
of  the  pier,  and  the  intersection  of  this  base  with  the 
curve  of  pressure,  that  is,  the  j^oint  of  application  of 
the  resultant  of  all  the  forces  which  act  upon  the  base. 

^=the  coefficient  of  stability. 

^=:the  unknown  thickness  of  pier. 

We  have  in  all  arches 

n-=^\r{^E~{-E')—i\iQ  curvilinear  surface  Ah  C^  figs.  4,  1*^. 

m—r'^{^E-\-\E'')—t}iQ  moment  on  A  of  the  surface  A  h  C\ 
figs.  4,  10. 
Let  the  resultant  pass  through  the  exterior  edge  of  the 

base,  a?=rO;  we  have 

\]ie^-\-ne-\-m=hFl ;  giving  (30) 


254  THEORY     OF    THE    ARCH. 

All  tbe  otlier  quantities  l)eing  known,  tlie  value  of  x  is 

x-^ '—f^. ;  (32) 

eli-\-n 

Let  the  resultant  ])as8  tlirougli  the  l)a.«e  at  J  its  length 
from  the  exterior  edge,  x—\e ;  and  let  fcl ;  we  have 

\c^li-^\ne-^m^Fl  (30i) 


Let  a?=|f^,  and  ^=1 ;  Ave  have 


Let  the  resultant  pass  at  any  proportional  distance,  pe^ 
from  the  exterior  edge,  x=-pe^  and  let  5=1,  we  have 

{l-j^y^Uj^{\-p)ne^Fl  (30/;) 


V  VI-2W      A'     r^j^;^  A  ^1-2^;^  A  ' 
Let  the  resultant  pass  through  the  middle  of  the  base, 
x^^\e^  and  let  5=1,  we  have 

\ne^m^Fl  (30^ 


e— 


-(^-^)  (^1^> 


The  values  of  <?  drawn  from  (31),  (3H),  (31|),  (31^^), 

dmci"  onlv  in  the  numerical  coemcients  01  -y,  tq?  -7-5  -7-;  ^^ 

A  /r   li     li 

that,  having  solved  one  of  these  equations,  we  can  readily 

solve  others. 

It  has  Ijeen  customaiy  in  large  arches,  to  assign  to  5, 

the  coefficient  of  stability,  the   value  of  1.90  or  2,   and 

determine  the  thickness  of  pier  from  an  e(piation  ecjuival- 


THICKNESS    OF    TIER.— GENERAL    FORMULAE.  255 

ent  to  (31).  But  this  is  not  alvvaj^s  enough.  We  may- 
still  want  to  know  where  the  resultant  of  the  thrust  and 
of  the  weight  of  the  semi-ai'ch  and  pier  cuts  the  base. 
This  is  given  by  (32)  for  any  assigned  values  of  h  and  e. 

When  we  have  determined  e  from  (30),  (31),  for  any 
particular  coefficient  of  stability  ^,  we  can  substitute,  in  the 
numerator  of  the  second  member  of  (32),  for  ^^^^-[-ne-^- 
771,  the  equivalent  (30),  SM,  which  gives  to  (32)  a  form 
more  convenient  for  computation,  viz. : 

When  5=2,  its  usual  value,  we  have  a?=-T 

Giving  to  h  and  I  in  (32)  the  particular  values  which 
correspond  to  the  springing  line,  we  learn  where  the  curve 
of  pressure  cuts  that  line. 

If  we  suppose  /,  the  lever-arm  of  the  thrust,  to  be 
variable  in  (31^),  e  also  being  variable,  that  formula 
becomes  the  equation  of  a  right  line,  very  easy  to  con- 
struct, whose  intersection  with  the  base  gives  the  middle 
of  the  pier. 

The  thickness  of  pier  should  never  be  less  than  that 
which  is  given  by  (31  J) ;  for  if  the  resultant  cuts  the 
joint  of  the  base  within  one  third  of  its  length  from  the 
exterior  edge,  that  joint  will  open,  or  tend  to  open,  at  its 
inner  edge. 

According  to  the  rule  deduced  by  Audoy,  from  an 
examination  of  the  magazine  of  Vauban,  we  suppose  the 
thrust  to  be  doul)led,  assume  S=2,  and  determine  e  from 
equation  (31).  This  rule  is  perfectly  safe  in  almost  all 
practical  cases;  but  it  gives,  to  piers  of  great  height,  a 
thickness  too  small,  by  causing  the  curve  of  pressure  to 
pass  within  the  proscribed  limit  just  mentioned,  and  by 
leaving  the  surfiice  of  the  base  too  small  to  bear  the  super- 
incumbent weight. 


256  TIIEORV     OF    THE    ARCH. 

To  pioi-s  of  small  height,  the  rule  gives  a  thickness 
uuiiece^sarily  large. 

Poncelet  recommends  as  a  rule  for  piers  of  small  height, 
that  tlie  resultant  should  pass  through  the  middle  of  the 
base.  The  required  thickness  is  then  given  by  the  very 
simple  formula  (31i),  which  we  here  repeat, 


=<^^> 


This  hast  rule  would  seem  to  be  a  good  one,  so  far  as 
the  pier  is  concerned,  provided  the  thickness  thus  deter- 
mined is  sufficient  for  the  superincumbent  weight;  l)ut 
Poncelet  did  not  intend  that  this  rule  should  be  followed 
blindlv.  It  will  often  give  results  much  too  small.  At 
the  springing  line  of  most  segmental  arches,  it  liecomes 
illusory,  for  there  we  have  Fl—m^  or,  by  the  rule,  f =0. 

If  we  knew  the  real  acting  pressure  at  the  crown  and 
its  point  of  application,  the  rule  would  be  perfect. 

To  construct  (31^) :  on  the  profile  of  the  magazine  of 
Yauban,  fig.  13,  from  «',  the  intei-section  of  the  inner  face 

of  the  i)ier  with  the  horizontal  a  a\  lay  off  a  u——  ;    on 

the  horizontal  a  a  prolonged  lay  off  n  i) —F^  and,  on  the 
vertical  tlirough  n^  lay  off  np=n.  The  diagonal  n  q  is  the 
required  line. 

It  is  easy  to  give  to  (31)  such  a  form  as  to  establish, 
in  its  utmost  generality,  the  princii)le  already  announced 
ioY  a  i)articular  case,  art.  43,  relative  to  the  limit  thickness 
of  pier.     Calling  i  the  difference  between  h  and  Z,  we  have 

h    ~~        h  h    ' 

Suppose  7i= infinity  ;  (31)  becomes  e=/25i'"'.  In  like 
manner,  we  find  tlu*  limit  tliickness  to  be, 

whenir=-!r^,  (31^),  e=\/^; 
"     x=^e,  {2>li),  e=\/WF ; 


THICKNESS     OF    PIER.— SEMICIRCULAR     ARCHES.  257 


wlien  x=.pe,  (Sl^)),  e=^ ^^-F ; 

"     x—\e^  (^H))  6= infinity. 

The  formuLTe  given  in  this  article  are  all  of  universal 
application,  whatever  be  the  load,  and  whatever  the 
curves  of  the  arch,  circular,  elliptical,  segmental,  &c.  But 
the  values  of  n  and  m,  given  above,  ai-e  based  on  the  sup- 
position that  the  surface  of  the  reduced  surcharge  over  the 
half-span  is  a  single  plane.  Should  that  surface  be  irregu- 
lar, it  is  only  necessary  to  say  that  n  is  the  sum  of  all  the 
surfaces  over  the  half-span,  and  m  the  sum  of  the  moments 
of  all  those  surfaces  in  reference  to  A^  figures  4,  10,  the 
interior  edge  of  the  joint  of  the  springing  line.  Should 
the  value  of  e  resulting  from  (31),  show  that  we  have  not 
estimated  li  very  correctly,  we  shall  be  able  to  correct  the 
estimate  and  calculate  e  anew.  Strictly  speaking,  h  is  not 
precisely  the  same  in  (31),  (32),  (SOJ),  tfec.  In  (32),  h  is 
j^roperly  the  mean  height;  in  (31),  fig.  10,  the  height 
measured  along  a  vertical  cutting  the  base  of  the  pier  at 
^  its  thickness  from  the  interior  face.  It  would  be  easy 
in  any  particular  case  to  make  the  proper  correction  ;  but 
such  correction  will  hardly  ever  he  necessary,  for  e  changes 
very  little  with  small  variations  of  h. 


THICKNESS    OF   PIEE. THE   INTEADOS    A    SEMICIRCLE. 

65.  The  formulae  of  the  preceding  article  all  remain 
unchanged:  but  we  have  (figs.  4,  10), 

the  curvilinear  surface  Ah  C,  =j7tr^=y"x  0.7854 

moment  on  J  of     "  "       rz:;-/^-- W;-^X0.4520G5 

Hence 

n=\r{E-\-E')-r''  x  0.7854 

w=?XJ-S'+i^')-''X  0.452065 


258  THEORY    OF    THE    ARCH. 

^and  E'  are  always  given  Ijy  the  conditions  of  the  prob- 
lem.    They  stand  in  this  relation,  E'^^E—v  cot.  I. 

If  the  roof  of  the  arch  he  inclined  45°  we  have  -E"= 
E-r. 

If  the  roof  is  horizontal,  we  have  E' —E ;  and,  intro- 

ducnisr  the  ratio  A  =  — , 


m__  /Ur-0.45-2065\ 


THE    ROOF    GREATLY    INCLINED,    AND    AVITII    LITTLE    OR    NO 

SURCHARGE. 

66.  When  the  roof  of  the  arch  is  so  steep  and  the  arch 
so  thin  that  we  can  regard  the  triangle  D  E  P^  tig.  13,  as 
forming  a  part  of  the  semi-arch  or  pier,  we  can  give  to  m 
and  n^  in  the  equations  of  art.  64,  a  meaning  which,  without 
changing  the  form  or  purport  of  any  of  those  formulae, 
shall  render  their  apjJication  somewhat  easier. 
Let  ?i'— the  surface  of  the  whole  semi-arch  and  its  reduced 

load,  and  of  that  part  of  the  pier  which  lies  above  the 

springing  line. 
7/i'=the  moment  of  that  surface  in  relation  to  the  interior 

edge  of  the  joint  of  the  springing  line. 
A'=the  height  of  the  pier  from  its  base  to  the  springing 

litie.    Let  E,  I,  F^  ?•,  a?,  5,  e  T)e  the  same  as  in  art.  64. 

"We  have  n=\  tang.  7x /i''-/' X 0.7854  ; 

7//  =  ^  tang.  7x^^X/'-J  tang.  /x^^-^'X ^-452065 

If  the  roof  be  inclined  45°,  we  have 

^/^^/X/r^-JiT^X  0.4714-0.452065). 


THICKNESS     OF    PIER.— THE     ROOF    STEEP.  259 

To  determine  the  thickness  of  pier,  we  have 

U'e''+7ie-{-m=SM  (30)' 


n'  ,      /?i''     2m'  ,  26FI  .^.y 

in  which  6  is  ordinarily  taken  at  1.90  or  2,  and  the  result- 
ant of  the  thrust  thus  increased,  and  of  the  weight  of  the 
semi-arch  and  pier,  passes  through  the  exterior  edge  of  the 
base. 

The  point  where  the  resultant  of  the  true  thrust  and  of 
the  weights  just  alluded  to  cuts  the  base,  is  given  as  fol- 
lows : 

^= eJ^-^' (^^') 

in  which  we  substitute  for  e  the  value  determined  by  (31), 
or  other  values  accordins;  to  circumstances. 

If  we  wish  to  determine  the  thickness  of  pier  on  the 
condition  that  the  resultant  of  the  true  thrust  and  of  the 
weight  of  the  semi-arch  and  pier  shall  intersect  the  base  of 
the  pier  at  a  distance  equal  to  one-third  its  thickness  from 
the  exterior  edge  (x=:^e),  we  have 

^eyi-{-f?ie+m=Fl  (30|)' 


If  .T=|6  we  have, 

i^(?K^-^e^m^Fl.  (30|)' 


n   .      I  jn  "     ^  „m    .  .  ^Fl 


,=  _-+V9p-10^  +  10^f.  (311)' 

\ixr=^pe^iy  being  any  fraction  whatever,  we  have 

(^_^;),///+(l  -jyye^m^Fl;  (30;;)' 


2C)0  THEORY    OF    THE    ARCH. 

Finally,  if  the  resultant  pass  tlirougli  tlie  middle  of  the 
base,  x=y,  we  have 

^n'e-\-m'=Fl  (30^)' 

\fi       n  J 

When  we  have  determined  <?,  (30)',  (31)',  for  any  par- 
ticular value  of  5,  we  can  give  to  (32)'  the  following  more 
simple  form : — 

ell  -\-ii 
which,  when  ^=2,  becomes  x 


eh'-\-ri' 

67.  Example  1.      The   magazine-arch,  with   the   roof 
inclined  45",  without  surcharge. 

^•=10';    i?=12';    7r=:^=--1.20;    /=-45";    n\  art.  66,  = 

65.46;  m\  art.  66,  =  173.356  ;  //  =  10';  l^li  ^R— 
22'. ;  F,  table  C,  =  r^x  0.25806=25.806  ;  from  these 
data  we  have,  by  calculation, 

^=6.546;  (  ^,)  =42.85  ;;==17.3356;  ^=56.7732;' 

hence,  l)y  the  formulae  of  art.  66,  for  strict  equilibrium, 

(5  =  1,  (31)',    e=:—6.54G  +  4/42.85  — 2x17.3356  +  2  x5G.7T32  =  4'.487. 

for  (5  =  2,   (31)',  e=  —  G.540  +  /42.85  — 2x17.3356  +  4  x5G.7732  =  8'.792. 

"  x=\e,{2,\\),  e=  —  2  x  6.546  +  1/4  x  42.85  —  6  x  17.3356  +  6  x  56.7732 

=  7'.107. 


"  x=lc,  (31|)>=  — 3x6.545  + V'9x42.85  — 10  x  17.3356  +  10  x  56.7732 

=  8'.29. 
"  x=\€,  (31^)',  c  =  12'.05. 

We  may  infer  without  further  calculation  that  the  rule 
of  Audoy,  which  consists  in  doubling  the  horizontal  thrust, 
or  assuming  (5  =  2  in  (31)',  is  in  this  case  perfectly  safe ;  for 


THICKNESS     OF    riER.-EXAMPLES.  2G1 

it  gives  a  greater  thickness  than  (31|)',  which  requires  the 
resultant  to  pass  ^e  from  the  exterior  edge. 

For  e=S'.1d2  we  find,  by  (32)',  a;=zS'.1=exOA21. 
Making  7i=0  in  (30)',  we  have,  for  strict  equilibrium,  or 
fcl,  t^=2'.08  ;  for  fc2,  ^=6'.81. 

By  comparing  (30)'  with  (31|)',  we  see  that  the  thick- 
ness required  for  strict  equilibrium  at  the  springing  line  is 
precisely  half  the  thickness  required  for  the  resultant  to 
pass  through  the  middle  of  the  base. 

Example  2.  The  magazine  at  Fort  Jefferson,  of  which 
we  gave  the  thrust  in  art.  63,  fig.  12  :  r=14:' ;  Ii=-[T.50  • 
X=:1.25;  7=::56°3'23";  t=  depth  of  surcharge  above 
the  tangent  planes  =  5'.90  ;  A=16'.5  below  the  springing 
line  +  13'.5  above,  =30;  /=16'.5  +  17'.5  =  34';  i^=:74.86; 

-^=g-i^+^=27';  £J'=i;-rcotI=ir.5S  (arts.  64,  Go)  ; 

7^.1=158.12  ;^;z=:1097.82;  ^'=6.943;  j=5.2l  ;  ^^=27.78; 

m  Fl 

^==36.594;  -^  =  84.841. 

With  these  data,  the  formulae  of  art.  64  give  us,  for 
strict  equilibrium,  or 

(J  =  l,  (31),  e=—5.27  +  '/27.V8  — 2x30.594  +  2  x84.841  =  5'.88. 

For  <J  =  2,  (31),  <?=— 5.2 7 +  1^2 7.78  — 2  x  36.594  +  4  x  84.841  =  11'.88. 

"   x—\e,  (311),  e— —  2  x  5.27  +  |/4  x  27.78  —  6  x  36.594  +  6  x  84.841 

=  9'.48. 

"   x—%e^  (31|),  c=— 3  X  5.27  +  1^9x27.78  —  10  x  36.594  +  10  x  84.841 

=  11'.26. 

We  see  that  the  thickness  required  by  the  rule  of 
Audoy,  11'.88,  is  amply  sufficient,  as  it  is  greater  than 
11 '.2  6,  corresponding  to  x^=.^e. 

But  the  pier  of  this  magazine  has  a  running  gallery  3' 
wide,  18'  high,  with  a  4'  solid  wall  outside,  and  with  its 
floor  nearly  on  the  level  of  the  base. 


0(;',0  THEORY    OF    THE    ARCH. 

As  tliis  case  may  occur  acrain,  we  subjoin  the  necessary 
moditicatioii  of  (30),  (31).  It  is  evident  that  the  pier  will 
not  go  over  in  a  solid  mass,  l)ut  that  the  divisions  each 
side  of  the  gallery  will  revolve  separately. 

Let  e  represent  the  thickness  of  the  solid  wall  ontside 
the  o-allery ;  e'  this  same  thickness  incretvsed  l)y  the  Avidth 
of  the  gallery ;  e,  as  usual,  the  unknown  whole  thickness 
of  the  pier  ;  li  the  height  of  the  gallery,  supposed  to  have 
its  floor  on  the  level  of  the  base. 

The  equation  of  moments  corresponding  to  (30)  is 

giving 


(n-}ie'\  ,       I ln-Kc'\     ^( 


In  the  present  case  we  have  6'=:-i';  6-"=e'  +  3'=T' ; 
A'=18'.  Substituting  these  values,  we  have,  for  strict 
equilibrium,  or  fcl,  t'=G'.59  ;  for  5=2,  ^=14'.04. 

Let  us  still  further  take  into  consideration,  the  effect  of 
mortar  upon  the  thrust.  The  arch  is  covered  with  con- 
crete so  as  to  make  the  vertical  joint,  extended  through 
this  concrete,  9'  long,  and  the  joint  at  the  reins  over  5'  long. 
We  suppose  c^  art  16,  to  be  25,  and  take  only  J  of  that 
effective  force.     We  must  subtract  from  the  thrust,  art.  49, 

(.7=0',   <I  =  r,\    y=7?-ir=10'.r,0),    \y^\^i(^--±^^ 

14.02,  which  reduces  jPto  60.84. 

Substituting  this  value  fori^^in  the  last  term  under  the 
radical,  we  have,  for  5  =  2,  6=11'.77.  The  actual  thickness 
ado})t(*d  for  this  magazine  is  12'. 

We  take  no  account  of  the  effect  of  adhesion  upon  the 
base  of  the  pier  ;  for,  in  consequence  of  the  division  of  this 
pier  into  two  parts,  and  of  the  great  length  of  the  lever 
arm  of  the  thrust,  this  effect  is  almost  notliing. 


THICKNESS    OF    riER.-EXAMPLES.  2G3 

Example  3.— An  arch  at  Fort  Porter,  fig.  11.  The 
thrust  of  this  arch  has  l)een  given  in  art.  03,  example  3. 
^•=6';  ^=:7'.668;  /r=  1.278  ;  7^^=22.03;  iLlS'  below 
the  springing  line  +  7'.6G8  al)OYe  =  25'.GG8  ;  /^r=18'  +  12'  = 
30';  i:=CIi'=U'.2S',  E\  on  the  side  least  loaded,  = 
^f/  =  12'.59  ;  n,  arts.  64,  65,  =52.3356  ;  m,  arts.  64,  65,  = 
149.254  ;  hence,  by  calculation, 

-=2.852;  -=U445;    -=3.0435;    '|=4.9V5  ;  ^'=19.302.5. 

With  these  data,  the  formnljB  of  art.  64  give  ns. 
For  strict  equilibrium,  ^=1,  (31),  ^=:3'.90. 

For  ^=2  (rule  of  Audoy),        "  f=rG'.65. 

Fora^=i^,  (3H),  ^=6'.44. 

For  ^=|6,  (31|),  ^=:7'.86. 

The  thickness  given  l)j  the  rale  of  Audoy,  6'.65,  is 
barely  sufficient,  as  it  exceeds  but  little  the  value  of  e  cor- 
responding to  a?= J(^. 

But  the  pier  of  this  arch  is  not  solid.  It  has  counter- 
forts on  the  inside,  3'  wide  and  6'.50  apart,  united  by 
arches  at  top  starting  from  the  level  of  the  springino-  line 
of  the  main  arch.  The  pier  outside,  4'  in  thickness,  is 
continuous  and  solid. 

Let  e'=ithe  known  thickness  of  the  solid  wall  on  the 
outside ;  ^=the  proportion  of  the  vacant  space  between 
two  counterforts  to  the  same  distance  increased  by  the 
width  of  one  of  the  counterforts  ;  A'=the  mean  heio-ht  of 
the  vacant  spaces  between  the  counterforts. 

The  equation  of  moments  corresponding  to  (30)  is 

y,e^j^ne+m—lpJi'(/-e")=hFI,  giving 


<?= 


n  (     n     Y     2m+yi'e'^,    nFl 


h-2)h'^y  \h-ph'j     ii—pK  '^h-ph' 


(34) 


In   the    present   case  we   have  p)^^=: — ;    //ml 9' 

nearly  ;  6'= 4'.    We  have,  therefore,  for  strict  equilibrium, 
6 


204  THEORY    OF    TUE    ARC  [I. 


or  ^=:1,  (:=:3'.85;  and  l>y  tlie  rnle  of  Audoy,  ^  =  -,  <:  = 
7'.71.     In  (34)  t  is  supposed  to  exceed  t . 

Example  4. — The  magazine  of  Vaul)an,  fig.  13. 
7'=12'.oO;  i4'=15'.50;  K—\.1\\  Z'=altitude  of  ridge 
above  the  springing  line=:20'.50;  /=40°  7'  17";  A'= 
8';  Z'=7i'+i?=:23'.50;  the  thrust,  already  given  for 
this  arch,  arts.  48,  G3,  is  i^=r-X  0.229381  =  35.84. 
From  these  data,  referring  to  art.  6G,  we  have  l>y  cal- 
culation, w'=  120.039  ;  w'  =  235.06;  i^^'  =  842.26. 

;/'  ;/'  in'  Fl' 

-  =  1.1005  ;  -;-7,=225.14G3  ;  7t=29.3825  ;  -yr=  105.2823. 

It  ^  If  ^  k  lb 

"With  these  values,  the  formulae  of  art.  66  give  us, 

for  strict  equilibrium,  c^  =  l,  (31)',  ^=4.410. 

For  ^=2,  rule  of  Audoy,  (31)',  <?=9'.234. 

For  x=\e,  5=1,  (3H)',  ^=6'.814. 

For  .T=|<?,  fcl,  (31|)',  6r=7'.761. 

For  x^\e,  5=1,  (SU)',  6=10'.117. 

When  <?=9'.234,  rule  of  Audoy,  (325)',    .r=:4'.34=cX0.47. 
AVe  shall  f^ive  a  discussion  of  this  arch  hereafter. 


SECTIOX  IV. 

ARCHES    IX    SEGMEXTS    OR.    PARTS    OF    A    CIRCLE,    USUALLY 
CALLED    SEGMENTAL    APvCIIES. 

68.  These  arches  are  very  common  in  fortifications,  and 
still  more  common  in  bridges  of  large  span. 

Indeed,  the  semi-circular  arch  of  large  sj^an,  and  of  the 
usual  thickness  at  the  key,  Avhich  is  about  -^-^  of  the  span. 
Las  a  great  tendency,  after  the  removal  of  the  centering, 
to  settle  down  at  the  key  and  spread  out  at  the  reins 
about  60°  fr(»m  the  key,  so  that  such  arches  can  only  be 
safely  used  "wlicn  their  thickness  is  greatly  increased  below 
tlu^  reins,  or  when  tlieir   piers   are   continued  al)ove   the 


SEGMENTAL    ARCHES.  205 

springing  line,  in  solid  and  almost  incompressible  masonry, 
as  liigli  as  tlie  reins. 

Sucli  arrangements  iu  effect  reduce  semi-circular  to  seg- 
mental arches. 

Segmental  arches  are  fully  given  when  we  know  tlie 
span=r,9,  the  rise  of  the  intrados  above  the  springing  line 
=/,  and  the  thickness  at  the  'k.ej=d. 

Let  r,  as  usual,  represent  the  radius  of  the  intrados,  v' 
the  half-angle  at  the  center.     We  have 


^'        '        '       s  ,  f 


*-i/+-^;  ^m.v=~;  cos.v'ml-Z. 

As  the  thickness  of  the  arch  at  the  key  is  given,,  we 
know  the  value  of  the  ratio  of  the  two  radii, 

r  r 

When  not  otherwise  mentioned,  w^e  shall  suppose  the 
thickness  of  the  arch  to  be  the  same  throughout.  Should 
the  thickness  increase  towards  the  reins,  the  formulas  and 
the  tables  to  be  explained  hereafter  will  give  a  slight 
excess  of  thrust. 

SEGMENTAL    ARCHES     WITHOUT    SURCHAEGE, INTRADOS    AND 

EXTEADOS    PARALLEL    FIG.    14. 

69.  Look  in  table  A  for  the  angle  of  rupture  corre- 
sponding to  the  given  value  of  K.  If  that  angle  be  less 
than  'v\  the  thrust  is  evidently  given  at  once  l)y  the  table. 
But  if  the  angle  v^  be  less  than  the  angle  of  rupture  in 
table  A,  it  is  easy  to  see  that  the  prism  of  maximum  thrust 
extends  to  the  springing  line.  The  thrust  (rotation)  will 
in  this  case  be  given  at  once  by  (11)  art.  28,  when  we  have 
substituted  for  v  in  that  formula  the  known  value  of  v . 

In  like  manner  the  sliding  thrust  will  be  given  1  )y  table 
A  when  v  exceeds  26°. 

If  y'  be  less  than  26°,  this  thrust  will  l)e  given  by  (17), 
art.  36,  Avhen  we  have  substituted  for  v  in  that  formula 
the  known  value  of  v' . 


2(3i5  TIIEURY     (»F     THE     ARCH. 

Tal»le  E,  calculated  l)y  M.  Petit,  give^,  for  all  tlie  val- 
ues of /r  between  1.01  and  1.40  inclusive,  the  actual  thrust 
in  seven  systems  of  segmental  arch,  l>eing  the  varieties  in 
most  common  use.  These  varieties  are  as  follows  :  ■s  =  4,  5, 
C,  7,  8,  10,  and  IG  times/'. 

Above  the  hoi'izontal  line  in  each  colunm,  the  sliding 
exceed  the  rotation  thrusts,  and  the  former  only  are  given. 
Below  the  horizontal  line  the  rotation  .thrusts  only  are 
given. 

If  the  angle  of  rupture  in  table  A,  corresponding  to 
systems  not  given  in  table  E,  that  is,  to  segmental  arches 
of  which  the  half  span  is  less  than  four  times  the  rise, 
exceed  v'  by  only  six  or  eight  degrees,  the  thrust  may  still 
be  taken  from  talde  A  without  sensible  error. 

Illustration. — Second  column  of  table  E,  -5  =  4/,  v'— 
53"  7'  30". 

For  7l  =  1.18  table  E  gives  i^=;-2x  0.10313 

"         "  "     A     "    r=58° 40',  i^=:/-X 0.10417 

Difference  in  the  angles,  v—v'  —  5°  32' 
30"  ;  error  in  the  thrust,  always  in 
favor  of  stability,  r-  X  0.00104 

70.  Tlie  7'otation  tlini-^t^  <luainifiliedhj  mortal^  is 

in  whicli  ^/'  =  tlie  thickness  of  the  arch  at  the  springing 
line,  ?--6'=the  thrust,  without  adhesion,  obtained  from 
table  E  or  by  direct  calculation. 

But  if  the  thrust  has  been  taken  from  table  A,  that  is, 
if  V  be  nearly  equal  to  v  or  exceed  i\  the  effect  of  mortar 
and  nf  surcharge  has  already  been  given  in  the  discussion 
of  the  semicircular  arch,  art.  31  and  following. 

71.  The  sliding  thi-ust  diminished  by  the  adhesion  of 
moi-tar  is,  ai-t.  37, 

F=r'^C—^l^^^^l-  (36) 

sm.  (t/+30°)  ^     ^ 


SEGMENTAL    ARCHES     WITHOUT    SURCHARGE.  20  7 

r^Ohemg  the  tlini>*t  obtained,  divectly  or  by  pro])()i'tional 
jDarts,  from  table  E,  or  by  an  independent  calculation. 

This  last  formula  is,  of  course,  to  l^e  used  only  when 
the  dimensions  of  the  given  arch  point  to  a  decimal  above 
the  horizontal  line  in  one  of  the  columns. 

li  v'  exceed  26°,  the  sliding  thrust  as  affected  by  mor- 
tar and  surcharge  has  already  been  given :  art.  37  and  fol- 
lowing. 

THICKI^ESS    OF   PIER. 

72.  Let  ??=: surface  of  semi-arch  a  h  m  n  a,  fig,  14  ; 
"    ^?i==  moment  of  that  surface  in  relation  to  m^ 
"      Zz^the  lever  arm  of  the  thrust,  or  elevation  of 
a  above  the  base  of  the  pier ; 
Let  7i=the  mean  known  or  estimated  height  of  the  pier 
from  its  base  to  the  upper  surface  of  its  surcharge  ; 
"  i<^=tlie  horizontal  thrust  however  determined ; 
"  i2?=the  distance  between  the  exterior  edge  of  the  base 
of  the  pier  and  the  point  where  that  base  is  crossed  by 
the  curve  of  pressure  ; 
"  ^=zthe  coefficient  of  stability ; 
"  (?=the  unknown  thickness  of  pier. 

We  suppose  the  small  triangle  m  n  a'  to  belong  both 
to  the  semi-arch  and  pier ;  thereby  greatly  simplifying  the 
formulae,  while  the  very  slight  resulting  error  is  always  in 
favor  of  stability.     We  have, 

When  the  angle  of  rupture  extends  to  the  springing 
line,  111  and  F  stand  in  this  relation. 


*  arc  of  l"=0.01'745i-;  lug.  of  dittos— 2.241S77  ;  r=\. 

It  will  be  most  convenient,  in  calculating  the  value  of  n,  to  expi-ess  «'  iu  degrees 
and  decimals  of  a  degree. 


2CS 


THEORY    OF    THE    AKCH. 


SO  that,  knowing  one  of  tliose  quantities,  we  can  ol>taiu  tlie 
other  without  a  sejiarate  calcuhition. 

The  subjoined  t'onnula^  are  identical  in  form  witli  thoi^e 
of  art.  04,  and  only  ditler  in  the  values  of  n  and  m^  which 
we  liave  given  al>t>ve. 


x=o 


J  ^ 


x= 


n^     2m     2SF/ 
Ue-  +  ve-^m-Fl 


t^=-^+Vp-T+'^^ 


eh  -{-n 


(32)^ 


AVhen  e  has  been  oljtained  from  (ol)/^,  for  any  par- 
ticular A'alue  of  ^,  we  have 


eh-\-)i 


AVhen  h=2, 


■/' — 


Fl 


ih-^ii 


Let  the  resultant  |»ass  through  the  l>ase  so  as  to  make 
x=^^e^  we  have 


no — L^ 


II  Fir-       in   ,     Fl 


lU—^eA 


^ne+m  —  Fl 
If  x=^e, 


\--(5-t) 


(30i)>9 

(3H).S' 
(30|).^ 

(3H)>S' 


The  discussion  of  these  equations  given  in  articles  G4, 
Go,  GG,  need  not  be  liere  i-ej)eated. 

It  is  necessary  to  determine,  once  for  all,  in  every  arch, 
tlie  values  of  F^  m,  and  //.  That  done,  the  above  equa- 
tions are  solved  Avith  ixreat  ease. 


THICKNESS    OF    PIER.— EXAMPLES.  2  GO 

73.  Example.  We  will  take  the  case  reported  l)y  Mv. 
Haiipt,  ill  his  very  excellent  work  on  Bridge  Construction, 
page  130.  "The  Monocacy,  a  very  violent  stream,  is 
crossed  by  a  beautifid  stone  bridge  (aqueduct),  of  nine 
arches,  each  54  feet  span,  and  9  feet  rise;  arches  2|-  feet 
thick,  abutments  10  feet  thick  and  10  feet  hiirh,  on  a 
foundation  3  feet  high  and  13  feet  wide. 

"  Some  arches  and  piers  had  been  built  up  and  backed 
in ;  but,  before  the  whole  could  be  completed,  a  great  flood 
swept  the  last  center  from  under  the  arch  just  turned  and 
not  backed  in,  except  partially,  on  one  side.  The  rise  of 
this  arch  being  only  one  sixth  part  of  the  span,  must  have 
pressed  with  tremendous  effect  upon  its  last  jiier,  especially 
as  the  supports  were  very  suddenly  knocked  from  beneath 
it,  and  it  w^as  brought  to  bear  very  suddenly  upon  the 
pier.  This  had  been  well  built  with  hydraulic  cement  of 
toleraljly  good  quality,  only  eight  or  ten  months  before. 
The  arch  stood  triumphantly,  contrary  to  the  expectation 
of  all  that  witnessed  it,  who  looked  for  nothiuo^  but  the 
destruction  of  every  arch  then  built,  one  after  another." 
The  pier  had  "  lost  much  of  its  specific  gravity  by  immer- 
sion." 

We  have  in  this  case  .5-=54' ;  /=:9';  d=2'.50  ;  r=:45'; 

i?=47'.50;  i''=36°52'10";  7r=l-f--=1.05555;   7^  =  10'; 

Z=21'.50;    6'=(if. 

This  arch  belongs  to  one  of  the  systems  of  table  E  (see 
column  4). 

The  thrust  given  in  that  column,  is, 
for  /i"=:1.06,  7^  =  ^^X0.04280 
"  /f=1.05,  i^=r'x0.03709 

r-X  0.03709 

^•2X0.00571X1        =y'x0.00317 


Hence  for  /r=1.05|  i^=?-X 0.04026 

:81.5265  n  =  14:Al;  m=F(^-\-2.5)  =  d37. 554:1 


270  TUEORY    OF    THE    ARCH. 

The  exact  values  of  m  and  F'  are  a  very  little  larger 
than  those  obtained  above  l)y  interpolation,  Ijut  the  differ- 
ences AVonld  produce  no  sensible  effect  ujDon  the  results. 

AVe  have,  l»y  calculation  from  the  above  data, 

^=7.441 ;  ^!=55.3T  ;  ^=93.7555  ;  ^'^=175.2820 
h  /r  It  h 

Substituting  these  values  in  the  forniuhx?  of  7*2,  we 
have,  fur  strict  etpiilibrium,  or 


8  =  \,{^\)S,    f=  — 7.44  +  1   55.37-2  X  93.7555  +  2  X  175.'282  =  '7'.34. 
For  : — 


(5  =  2,         "      c=:- 7.44  +  1/55.37  —  2  X  93.7555  +  4  xl75.282  =  lG'.41. 

«5=1.25,    "      f=  — 7.44  +  /55.37  — 2x93.7555  +  2|xl75.282  =  10'.05. 

.r  =  ^c  (31  J)5',     e=  —  2  x  7.44  +  ^4  x  55.37  — G  x  93.7555  +  G  x  175.282  = 

11'.78. 


.r=|e(31|).S,Ci^  — 3  x  7.44  +  ^9  x  55.37  —  10  x  93.7555  +  10  x  175.282 

=  13'.92. 

.r  =  ^e(3U),S,  ,  =  2/---j=21'.91. 

Had  the  pier  lost  one  half  its  weight  by  immersion,  we 
find,  substituting  \lie^  for  ^lit"  in  (30)>S;  the  thickness 
necessary  for  strict  ecpiilibrium  to  be  only  8'52. 

AVe  learn  from  (32)6',  that  the  resultant  of  the  thrust 
and  of  the  Aveight  of  the  semi-arch  and  i)ier  crossed  the 
base  at  the  distance  ir=:2'.4G=t'X  0.246  from  the  exterior 
edge.  Consequently,  the  foundation-joint  of  the  pier  was 
open  on  the  inside  as  far  as  (10'-3x2.'46)  =  2'.G2  from 
the  inner  edge. 

We  have,  in  fact,  overrated  the  staljility  of  this  pier; 
fur  the  til  rust  given  in  unr  tables  is  the  horizontal  pi*essure 
acting  at  the  crown  of  the  arch  at  the  moment  of  rupture, 
and  is  not  so  great  as  the  existing  pressure  where  the 
thickness  of  jjier  is  such  as  to  prevent  rupture.  It  is  inter- 
esting to  remark  that,  had  the  opposite  half  of  this  arch 
been  loaded  in  masonry  up  to  the  hoiizontal  tan,<i:ent  to 
the  extradosat  the  crown,  the  thrust,  table  E',  would  have 


SEGMENTAL    ARCHES    SURCHARGED     IIOlllZOXTALLY.        2^1 

been  increased  fifty  per  cent.,  ^^'llile  the  elements  of  resist- 
ance would  have  remained  the  same.  Consequently,  the 
pier  would  have  been  overturned,  for  we  have  found  that 
all  increase  of  twenty-five  per  cent.,  ^=1.25,  recjuired  a 
thickness  6=10'.05.  Had  the  surcharge  been  only  one 
half  as  heavy  as  masonry,  the  pier  would  have  been, 
^=1.25,  almost  exactly  in  equililjrium. 


SEGMENTAL   ARCHES    SURCHARGED    HORIZONTALLY. 

Figure  15. 

'74.  This  is  the  most  common  form  of  the  river  arch. 

The  surcharge  of  masonry  and  earth  usually  rises  to  a 
horizontal  plane  passing  a  little  above  the  extrados  of  the 
key. 

For  the  present  we  shall  suppose  this  horizontal  upper 
surface  to  be  tangent  to  the  extrados  at  the  key;  and 
we  shall  continue  to  suppose  that  the  load  between  this 
plane  and  the  extrados  is  of  equal  density  with  tlie 
masonry  of  the  arch. 

For  notation,  see  art.  68. 

Look  in  table  D  for  the  angle  of  rupture,  %\  corre- 
sponding to  the  given  value  of  K.  If  that  angle  be  less 
than  v\  the  thrust  is  given  at  once  1)y  the  table ;  and  the 
effect  of  mortar  and  of  surcharge  will  be  the  same  as  in 
semi-circular  arches,  arts.  14,  49,  50. 

But  if  V  be  less  than  v^  the  prism  of  maximum  thrust 
extends  evidently  to  the  springing  line ;  and,  F  as  usual 
denoting  the  rotation  thrust,  we  shall  liave,  after  substi- 
tuting the  known  value  oiv  for  f  in  (28) 

(j{IC—GOS,v)  i  ^  \  J  ) 

Zv 1 

siu.u'     cos^j/;' 
In  like   manner  the  sliding  thrust  will  be  given  by 


21 '2  THEORY     OF    THE    ARCH 

talde  D,  if  ^''  exceed  29°;  and  the  effect  of  mortar  and  of 
surcliarire  will  1)e  tlie  same  as  in  semi-circular  arches. 

But  if  v'  he  less  than  29°,  the  sliding  thrust  Avill  he 
given  hy  (25)  when  we  have  substituted  for  TJ  in  that 
formula,  90°,  and  for  v  the  known  value  of  v. 

Tahle  E',  calculated  for  this  paper,  gives,  either  directly 
or  hy  i)i-oportional  parts,  for  all  values  of  ^between  1.01 
and  1.40,  and  for  all  I'elations  of  the  rise  to  the  sjxan 
l)etween  s=4:f  and  ^=16/',  the  horizontal  thrust  at  the 
extrados  of  the  key. 

Table  E'  is  altoo^ether  analos^ous  to  table  E. 

Above  the  liorizontal  line  in  each  column,  the  sliding 
exceed  the  rotation  thrnsts,  and  the  former  only  are  given. 
Below  the  horizontal  lines,  the  rotation  thrusts  only  are 
given. 

Should  the  angle  of  rupture  in  table  D  exceed  v  by 
only  five  or  six  degrees,  the  thrust  may  still  be  taken  from 
that  table  without  any  sensible  error;  and  the  effect  of 
mortar  and  of  surcharge  will  be  the  same  as  in  semi- 
circular arches. 

75.  The  rotation  thrust  diminished  by  mortar,  is 

?-^ (7  being  the  thrust  obtained  fi-om  table  E'  or  by  direct 
calculation,  d  and  d  respectively  the  length  of  the  upper 
and  lower  joint,  both  extended,  if  we  ])lease,  beyond  the 
true  extrados  of  the  arch,  through  a  cover  of  masonry  or 
concrete.  If  there  1)e  no  such  cover  at  the  vertical  joint, 
we  have  i/=f-\-d.  In  all  cases  y—f-\-i\\Q  thickness  of  the 
arch  proper  at  the  key. 

21ie  TotatloR  tliru.s-t  increased  Jnj  a  surcliar(je  of  the 
constant  vertical  de|)th  t^  is 

or,  according  to  convenience, — 


/)■■'— (i 


F=r'C-^U  %m^v'yi'-^'\  (38)' 

ill  wliicli  o-^Ch  the  tlirust  given  by  table  E',  or  obtained 
by  direct  calculation,  t  tlie  constant  depth  of  the  siir- 
charsre  above  the  horizontal  drawn  tan<2:ent  to  the  extrados 
at  the  key,  d  the  length  of  the  vertical  joint,  cl  the  length 
of  the  joint  at  the  sj^i'ingiiig  liue. 

'76.  The  sliding  thrust  mcreased  ly  surcharge  and 
diminished  hy  mortai\  is 

cd!  cos  30° 
p^,^(7+^(r  +  .r)  sin.  V  cot.  (30°  +  r')--^7q::^  ',  (^9) 

r^Q  l)eing  the  thrust  obtained  from  table  E'  or  T)y  direct 
calculation.  This  formula  is,  of  coui'se,  to  be  used  only 
when  the  dimensions  of  the  given  arch  point  to  a  decimal 
above  the  horizontal  line  in  table  E'. 

\i  V  be  nearly  equal  to  25°,  or  exceed  25°,  the  sliding 
thrust  and  the  effect  of  surcharge  are  given  in  tal  )le  F ; 
the  effect  of  mortar  becoming  at  the  same  time,  art.  37, 
rc(yK—V)^  or  rather,  cd' . 

77.  It  generally  happens  that  segmental  arches  of  large 
span  increase  in  thickness  from  the  summit  to  the  spring- 
ing line.  In  such  cases  our  formulae,  and  the  tables 
founded  upon  them,  give  thrusts  a  little  in  excess,  for  we 
neglect  the  small  trapezoid  n  n!  r'  ?',  fig.  15,  whose  weight 
is  in  favor  of  stability. 

Table  E'  will  give  the  thrust  of  such  arches,  very 
slightly  in  excess ;  but,  in  estimating  the  value  of  K  for 

the  horizontal  column,  we  no  longer  make  A^=— ,  for  these 
radii  may  be  drawn  from  different  centers ;  but  we  have 

The  effect  of  surcharge  and  mortar  may  be  obtained 


274  TUEORY    OF    THE    ARCH. 

from  the  formnljo  of  arts.  75,  70,  wliicli  apply  accurately 
to  the  case.'=i  under  consideration. 

Those  who  wish  to  attain  entire  accuracy  have  only  to 
subtract,  from  tlic  thrust  t-C\  as  given  by  table  E',  the 
ft  »lluwing  expression : 

i  sin.^  V  ((P—d-)—\  sin.^  v  cos.  r'l  — -, — ^  ) 

T8.  Examj)le.  Casemate  arch  of  Fort  Jefferson,  sup- 
porting the  second  tier  of  guns.     Figure  16. 

The  data  are,  6  =  15';  /'=2';  ^/=1'.50;  ^7=rl'.50; 
from  which  we  deduce,  7'=rl5.06;  i"'=30°  nearly;  A'= 
1.10,  nearly;  and  ,y=7i/*.     Table  E' gives 

for  7r=1.10  and  .5'=i7/,     i^=?''x  0.06784 

"  7^=1.10  and  6  =  87^;     i^=/-'x  0.05967 

hence   "  7^=1.10  and  6=71/ -^=''x0.06375  =  U.4635 

A  direct  calculation  gave  F=ir  X  0.06390 

the  difference,  r'^X  0.00015,  is,  in  effect,  nothing. 

But  the  arch  has  a  surcharge,  6  inches  dee])  through- 
out, which  adds  to  the  thrust,  art.  75,  if  we  suppose 
v=2>0\  4.00987. 

By  way  of  illustration,  let  us  attribute  to  the  mortar  of 
the  arch  and  of  the  concrete  which  covers  it,  an  adhesive 
force  of  3000  pounds  per  square  foot.  The  masonry 
weighs,  say  120  pounds  per  cubic  foot ;  hence,  art.  16, 
<?=3_o_V=25.  We  have  7,  the  depth  of  the  arch  at 
the  key,  I'.SO;  and  7',  the  dei)th  of  the  arch  and  concrete 
at  the  springing  line,  a  little  over  4",  say  4'.     Substituting 

these  values,  we  have  as  the  effect  of  mortar -i-r/—; — r)~ 

21.72;  and  the  final  thrust  7^=14.4635  +  4.00987-21.72  = 

—  3.25.     That  is,  the  arch  has  no  thrust.     If  we  disregard 
the  effect  of  mortar  upon  the  vertical  joint,  we  have  7^= 

—  .57  ;  still  n<j  thrust. 


SEGMENTAL     ARCHES    fUKCIlArOID     KOrJZC  KTALLY.        275 
THICKNESS    OF    TIEll. 

79.  Let  6"=: the  span;  /"=tlie  rise;  <'7=tlie  tliickness  of 
the  arch  at  the  key ;  ^=the  depth  of  tlie  surcharge  above 
the  key,  the  upper  surface  being  horizontal. 
n=ihe  surfjxce  of  that  part  of  the  arch  and  its  load  which 

lies  directly  over  the  half -span ; 
^«=itlie  moment  of  that  surfoce  in  relation  to  the  vertical 

passing  through  the  interior  edge  of  the  joint  of  the 

springing  line  ; 
Z=the  lever-arm  of  the  thrust,  or  elevation  of  the  point  a 

above  the  base  of  the  j^ier. 
h=zthe  entire  height  of  the  pier  from  its  base  to  the  top  of 

the  surcharare  over  it^z^'D'  fics:.  15. 
i^  x,  ^,  (?,  the  same  as  in  art.  72. 

"We  have 

«=i,s-(/+t/-|-)')  — 1)°(2«'— sin.  2?/)  (40) 

=h'(f+'l+t)->'(i>y  sin.  ^■+^!:-A       (41) 

The  formulae  which  give  the  thickness  of  pier  under 
various  circumstances,  are  precisely  the  same  as  in  art.  72, 
and  need  not  be  here  repeated. 

Example.  The  lower  casemate  arch  of  Fort  Jefferson, 
regarding  the  floor,  eight  and  one  half  feet  below  the 
springing  line,  as  the  base  of  the  pier.  The  data  are, 
/^  =  12'.50;  .9=15';  /=2';  f^=l'.50;  /=0'.50;  r=15'.0G  ; 
t''=30°;  /r^l.lO;  /=12'. 

This   is  the   arch  of  T^hich  we   obtained  the  thrust, 
i^=  14.4(3  in  art.  78.     The  above  data  give  us 
??  =  30.-r2x0.04529=:19.73;    m  =  112.50-r'^X0.0141  = 
64.34. 

71  n^  m  FT 

7=1.58;  7^^=2.49;  -^=5.15;  -y- =13.88 

Hence,  for  strict  equilibrium,  ^  =  1,  in  (31)>S',  (?=2'.S9 

for  ^  =  1.40,  in  (31)>S; ^=3'.99 

''x=\^,{Z\\)S, ^=4'.74 


'}n 


27G  THEORY    OF    THE    ARCH. 

We  see  that,  disregarding  tlie  effect  of  mortar,  the  pier 
should  be  at  least  4'. 7 -4  thick. 


SEGMENTAL  ARCHES  WITH  A  SURCHARGE  ON  EACH  SIDE  OF 
THE  CENTRAL  RIDGE,  RISING  TO  A  PLANE  OR  ROOF  AS 
IN   THE   MAGAZINE   ARCH,    FIGURE    19. 

80.  Let  6'=the  span  ;  /'=the  rise  ;  ^=:the  depth  of  the 
surcharge,  if  any,  above  the  two  planes  pai-allel  to  the  roof 
and  tangent  to  the  extrados ;  -u'^the  semi-angle  at  the 
center  ;  ^'^tlie  radius  of  the  intrados  ;  ^/=the  thickness  at 
the  crown;  <^^'=:the  thickness  at  the  springing  line  ;  !=■ 
the  angle  between  the  roof  and  a  vertical.     We  have 

7-    .      '^  ,  ^  ,  -"^     -'       ,      ^  ,     -,      ^ 

Yl  =  1 H —  ;  r=  ^^t -[--—,'■,  sm.  v  =  --- ;  cos.  v  —1  — -. 

/•  -^       8/  2r  r 

Look  on  table  F  for  the  angle  of  rupture,  i\  corre- 
sponding to  the  given  values  of  -/iTand  I.  This  angle  is 
given  in  three  columns  only,  viz.,  under  7=90°,  60°,  and 
45°.  Its  value  for  other  values  of  /  may  l>e  estimated 
Avitli  sufficient  accuracy  T)y  inspection,  as  it  will  l)e  suffi- 
cient for  our  present  purpose  if  we  kn(»w  tliat  angle  within 
six  or  eight  degrees.  If  that  angle  be  less  than  v\  or  ex- 
ceed v  \)j  only  six  or  eight  degrees,  the  thrust  and  the 
effect  of  surcharge  are  given  at  once  by  that  table,  pre- 
cisely as  if  the  intrados  were  a  semicircle.  The  effect  of 
mortar  will  also  be  the  same  as  in  semicircular  arches. 

But  if  V  be  less  than  v^  the  angle  of  greatest  thrust 
extends  evidently  to  the  springing  line,  and  the  thrust 
itself  will  be  given  by  (24),  when  we  have  substituted  for 
V  and  7,  in  tliat  formula,  the  known  value  of  v'  and  /in  the 
given  arcli. 

In  like  iiiaiiiHT,  \i  v  exceed  say  25°,  the  sliding  thrust, 
if  greater  tlian  the  rotation  thrust,  will  ])e  given  by  taljle 
F ;  and  the  effect  of  surcharge  will  also  be  given  1  >y  that 
taljle. 


SEGMENTAL     ARCHES    WITH     INCLINED    ROOFS.  277 

But  if  V  be  less  than  25°,  the  sliding  thrust  Avill  Le 
given  by  (25),  Avhen  we  have  substituted  for  v  and  /  in 
that  ecjuation,  the  known  values  of  v'  and  /  in  tlie  given 
arch. 

The  rotation  thrust,  diminished  by  the  effect  of  mor- 
tar, is 

y 

r'^Chemg  the  thrust  obtained  from  taT)le  F,  or  by  direct 
calculation,  and  the  last  term  being  the  effect  of  mortar ; 
d  and  d'  may  be  the  whole  length  of  the  vertical  and 
lower  joints  extended  through  any  cover  of  masonry  or 
concrete.  When  there  is  no  such  cover  at  the  vertical 
joint,  we  have  y=:^f-\-d.  At  all  times  we  have  y=ir{^K— 
cos.  v'). 

The  rotation  thrust,  increased  by  a  surcharge  of  uni- 
form depth,  t,,  above  the  roof  of  the  arch,  which  last  we 
suppose  to  be  tangent  to  the  extrados,  is 

^•^C' being  the  thrust  independently  of  surcharge. 

The  sliding  thrust,  increased  by  surcharge  and  dimin- 
ished by  the  effect  of  mortar,  is 

F^r'C-^t{r+d')  sin.^'  cot.  (30°+ 1'')--.'''^'  ''''^'  ^^° 


sin.  (y'+30°) 

^■^(7  being  the  sliding  thrust  without  regard  to  surcharge 
or  mortar.  In  this  formula  we  suppose  v  to  be  less  than 
25°  ;  if  greater  than  25°,  the  sliding  thrust  is  the  same  as 
in  semicircular  arches,  and  is  given  at  once  by  taljle  F, 
whenever  the  sliding  is  greater  than  the  rotation  thrust. 

81.  Example.  The  upper  casemate  arch  of  Fort  Jef- 
ferson, figure  17,  upper  part ;  surcharged  with  concrete  up 
to  the  roof,  A  0,  and  above  that  roof  with  earth  up  to  a 
horizontal  line  8|-  feet  above  the  springing  line.  Tlie  data 
are,6•=15';/=3';^;=13°36';^— 10'.875;(/=:^-r=2'.28; 


:27S  THEORY   of  the  arch, 

/r=1.21;  relative  weights  of  equal  volumes  of  eai-tli  and 
masonry  as  3  to  4. 

Reducini?  tlie  elevation  of  the  snrcharore  of  earth  in  the 
proportion  of  3  to  4,  Ave  may  regard  all  below  the  reduced 
sui-fiice  I)'  B!  O  as  having  the  density  of  masoniy. 

Drawing  the  line  R  I)  parallel  to  B!  I)'  and  tangent 
to  the  extrados  of  the  arch,  we  divide  the  figure  of  the 
semi-arch  into  two  parts;  the  one  including  all  Ittdow  this 
tangent ;  the  other  a  surcharge  of  uniform  dei)th  above 
that  line. 

The  angle,  /,  between  R  D  or  Ft  D'  and  a  vertical,  we 
find  to  lie  82°  24'  20".  As  the  angle  of  rupture  in  table  F 
corresponding  to  7^=1.21  and  7=90,  is  63°,  and  the 
angle  corresponding  to  7^=1.21  and  7=60  Is  54°,  -we 
know  that  the  prism  of  maximum  thrust  extends  to  the 
springing  line. 

Substituting  for  K^  v\  and  7,  in  (24),  the  values  above 
indicated,  Ave  obtain  7'=y-X. 10727  =  12.687.  Table  F 
gives,  for  the  same  values  of  7^  and  7,  7'=/-^X  0.12141. 

The  addition  to  the  thrust  caused  ])y  a  surcharge  of  uni- 
form depth,  ^=2'.725,  is,  art.  80,  =  13.886;  giving  as  the 
entire  thrust  7^=26.573. 

The  effect  of  the  adhesion  of  mortar  upon  the  thrust  is 
(7=4';  7=4';  2/=.5.28;  ^=25.), 

x,^^!±i'Lo5  25- 

leaving,  as  the  final  tlnust, 

7^=26.573-25.25  =  1.32. 

In  assuminix  ^^=25,  we  have  not  over-estimated  the 
effect  of  good  mortar,  and  may  regard  the  arch  in  question 
as  withont  thrust,  j)rovided  there  be  no  cracks  in  any  of 
the  joints.  Unfortunately  such  cracks  are  very  apt  to 
occur,  even  during  the  construction  of  the  arch.  We  have 
supposed  the  vertical  and  low^er  joints  to  extend  into  the 
concrete  covering,  making  7  and  d'  each  4  feet. 


I 


SEGMENTAL     ARCHES.— THICKNESS     OF    PIER.  2*79 

THICKNESS    OF   PIER. 

82.  Let  ,5= the  span  ;  y=tlie  rise  ;  c/=:the  thickness  of 
the  arch  at  the  crown. 

.^r=:the  elevation  of  the  reduced  ridge  above  the  spring- 
ing line=:w/7?',  fig.  19. 
^'=:the  elevation  of  the  reduced  roof  above  the  springing 

line,  measured  on  the  inner  face  of  the  pier,  j^rolonged, 

=m  a\  fig.  19. 
w=:the  surface  of  that  part  of  tlie  semi-arch  and  its  load 

which  lies  directly  over  the  half-span. 
??i=the  moment  of  that  surface  in  relation  to  the  inner 

face  of  the  pier. 
Z=tlie  lever  arm  of  the  tlirust=ff  </,  fig  19. 
A=the  mean  height  of  tbe  pier  from  its  base  to  the  top  of 

the  reduced  surcharge  upon  it,  to  be  estimated  if  not 

known. 
^=:the  coefficient  of  stability.    i^=the  thrust. 
6'=the  unknown  thickness  of  pier. 
E  and  JE'  are  always  known. 

We  have 

n={s{E+E')-\r\'lv'-m^.2v').  (42) 

m-^-,^-,6\E-\-^^E')-r'Uv'  sin.  z;'+^--i|     (4P,) 

The  formulae  w^hicb  give  the  thickness  of  pier  under 
various  suppositions,  are  precisely  the  same  as  in  articles 
64,  T2,  and  need  not  be  repeated.  The  formulae  of  64, 
we  have  already  said,  are  universal.  The  reader  is  referred 
to  that  article  for  a  discussion  of  the  formulae,  and  for  the 
equation  of  the  curve  of  pressure  or  resistance  in  the  pier. 

Example.  The  upper  casemate  arch  of  Fort  Jeftei-son, 
of  w^hich  we  obtained  the  thrust  in  art.  81.  Let  the  floor 
10  feet  below  the  springing  line,  l)e  the  base  of  the  pier, 
fig.  17. 

We  have  E=zm!  FJ =^' .Uh  ]    E'  =  ma=r.V2^\  r= 
10'.875;i;'=:43°36'10";  -s^lb' ]  h^lT  ]  1=15' .2S  ;  f=^' ; 
1 


OgO  THEORY     OF    THE    ARCH. 

^/^Sr.lO-^-X 0.1308  =  41.73;  ;/^  =  -219.l4-/'x0.055G6  = 
1-17.55  ;  7^=20.573. 

^'  =  2.455;  '^,=G.02G;  ^  =  8.08;  4-=23.882. 
h  It'  It  It 

AVith  these  data,  the  forimiLe  of  art.  G4  give  u^j, 

For  strict  equilibrmm,  ^=1,  (31),           .         .  ^=3.58 

For  5=2,  rule  of  Audoy  for  large  arches,  (31),  c^=G'.72 

For  5=1.50,  (31), e=5'.31 

For  .T=i^,  (31i), ^=5'.83 

For.r=i^^,  (311), 6=7'.00 

For.r=if,  (3U), f=12'.40 

The  thickness  given  by  the  rule  of  Audoy  would  seem, 
in  this  case,  to  be  about  right ;  as  it  is  nearly  equal  to  that 
which  corresponds  to  a?=|6. 

Were  we  to  take  into  consideration  the  adhesion  of 
mortar,  and  give  to  that  force  one  half  the  value  assigned 
in  art.  81,  we  should  hud  the  actual  thickness,  4',  to  be 
ani})ly  sufficient. 

THRUST  OF  THE  COM:*IUXICATION  ARCHES  OF  A  FORT  UPOIN" 
THE  SCARP  WALL,  AND  THE  CURVE  OF  PRESSURE  IX  THE 
LATTER. 

83.  This  is  one  of  the  most  important  applications  of 
the  theory  of  the  arch.  The  scarp  should  l)e  al)le  to  re- 
sist the  thrust  of  the  communication  arches  without  any 
lateral  motion  whatever ;  and  to  this  end  the  curve  t>f  j^res- 
sure  in  the  scarp  should  pass  through  the  middle  of  the 
foundations,  or  very  near  that  point. 

Etich  communication  arch,  besides  its  own  proper  load, 
supp(jrts  througli  its  entire  span  the  weight  of  one  half  of 
each  of  the  adjacent  casemate  ai'ches,  Mith  all  the  surcharge 
of  earth  and  masonry  which  may  l)elong  to  tlie  lattei*. 

The  thrust,  therefore,  of  the  communication  arches,  and 
particularly  of  tlie  upper  one,  is  very  great ;  and  the  effect 
of  the  latter  is  still  further  increased  by  tlie  great  leverage 


STABILITY     OF    SCARPS.— EXAMPLE.  281 

with  wliicli  it  acts,  that  is,  by  its  great  elevation  above  the 
base  of  the  scarp. 

These  arches,  on  the  outside,  rest  upon  small  piers  car- 
ried up  in  contact  with  the  scarp  wall,  but  nowhere 
bonded  in  with  it. 

84.  Example.  Communication  arches  and  scarp  of 
Fort  Jefferson,  figs.  17,  18. 

Lower  arch.  Sjxan = *  =:  1 2 '. 2  5  ;  rise  =/=  1 '.  7  5  ;  tliick- 
ness  at  the  key=6/=l'.88 ;  depth  of  surcharge  above  the 
key=^=:2'.87  ;  radiusof  the  intrados=:r=ll'.59  ;  K=l^ 

-=1.161;  6'= 7/;  elevation  of  the  extrados  of  the  crown 

above  the  base  of  the  scarp=/— :11'.63  ;  surface  711  h  m  of 
the  segment  of  the  adjacent  casemate  i!ivc\\=:a^=\r'*^{2v'  — 
sin.  2y')  =  20.54. 

But  we  advise  the  reader,  in  all  problems  of  this  kind, 
to  regard  the  segment  of  a  circle  as  the  segment  of  a  para- 
bola standing  on  the  same  span,  and  tangent  to  the  circle 
at  the  summit.  This  greatly  diminishes  the  labor  of  the 
calculation,  without  leading  to  any  sensible  error  in  the 
results.  According  to  this  supposition  we  have  <3f=-|  span 
Xrise=:(in  the  case  presented  above)  fx  15x2  =  20. 

The  center  of  gravity  of  the  semiparabolic  segment 
standing  on  a  horizontal  base,  is  at  the  distance  of  -|  of  this 
base  from  the  altitude  or  axis  of  the  whole  segment. 

Let  F  as  usual  represent  the  thrust,  we  have 

i^=  4/' X 0.0832.  .table  E',   the  thrust  without 

surcharge, =43.10 

4-4xiX7rT—^.  .effect   of    surcharge    trom   ref 

j-\-d 

(17'.13)  to  ref  (20'),  ....  =59.34 

4x15  — «     'S'^ 

X  — ..effect    of    the  adjacent   case- 


f+d  8 

mate  arch  from  (16')  to  (20'),     .         .       =206.70 

i^=309.20 


232  THEORY     OF    THE     ARCH. 

We  liave  computed  the  thrust  upon  the  whule  length 
(»t'  one  l)iei',  whicli  is  4  . 

In  computing  the  surcharge  we  suppose  the  load  to  be 
limited  tu  the  inner  face  of  the  pier,  and  not  as  usual,  to 
extend  over  the  skewback. 

Upper  arch  5  =  12'.25  ; /=3' ;  ^7=1'.88;  ^=12';  r= 
7'.7527;  /r=1.24i-;  6  =  4.08^/';  /=25'.63;  ^7=|15x3 
=  30. 

i^'=:4/-X 0.13233.  .table  E',  thrust  without  sur- 
charge,   .......     =31.81 

_^4  X  i  X  -A-f . .  effect  of  surcharge  from  (31'.13) 

to,  say,  (43'.13),      .         .         .         '         .  =184.50 

_Ll^lin^X  —  .  .effect   of   adjacent   casemate 
+      f^d  8  '' 

arch,  (30')  to  (43'),         ....  =634.23 

i^'=850.54 

Let  ^=the  volume  of  all  the  solids  between  the  inte- 
rior face  of  the  scarp  and  the  parallel  vertical  plane  pass- 
in£r  throu<di  the  crown  of  the  communication  arches  and 
ab^jve  the  iloor  of  the  lower  casemates,  which  is  at  the  ref- 
erence (7.50). 

J/=the  sum  of  the  moments  of  these  solids  in  refer- 
ence to  the  interior  face  of  the  scarp. 
JV=4  X  2  X  37.50 .  .  the  i)ier   pro])er  from 

(5.50)  to  (43),       ....  =300.00 

(10  05  12  25  \ 

0.50xi^'-tX-^Xl.75J      the 

lower  arch  from  (13'.50)  to  (20),      .      =130.G7 

/              12.25  12.25     .A  ^, 

+4hG.75x-^ |X-Y-X3Jthe  up- 
per arch  and  load,  (26'25)  to  (43'),         =301.36 

Carried  t..  page  283,         .         .         .     792.03 


STABILITY     OF    SCARP?.— EXAMPLE. 


283 


Brot.  forward, 
+  (15X4-|15X2)A  + 


792.03 


12.2;' 


lower  case- 


mate arch  and  load,  (16')  to  (20'), 

,  /       12.2rA 
+  (13X15-115X3)    2+-^,-       upper 


=  325.00  (IV) 
casemate  arcli  and  load,  (30')  to  (43'),  =1340.63     (V) 


iV^=2457.66 
31—2  X  4  X I X  37.50,  the  pier  from  (5'.5)  to  (43'),  =300.00 

r  ,,  12.25/       12.25 \,  1 

I  4(6.50  X  ^-(2+-^-))  I 

+  ^  loor;  /      •  io,on\     (^   l^Wer 


12.25         ^^        /     •         12.25\  I 
-4X-^-X1.75x|   2-i-IX^— 


+ 


communication  arch  and  load  from  (13'. 50) 

to  (20'), =639.64 

^      ^         12.25/^  ,  12.25' 
4x(16.75)X-2-(2+-^ 


-4X|X 


12.25 


X3   2-f-l-x 


12.25 


upper 


=  1791.97 


b 


com.  arch  from  (26'.25)  to  (43'),       . 

/       12.25\^ 
-f  (IV)  325  X  M  2  + — ^—  I  lower  casemate  arch 

from  (16')  to  (20'), =1320.31 

/       12.25\ 
+  (V)  1340.63  X  i(  2 H ^^—  I   upper  casemate 

arch  from  (30')  to  (43'),   ....     =5446.31 

i/=:  9498.2  3 

We  have  calculated  the  values  of  F,  F\  N,  and  J/" for 

19'  in  length  of  the  scarp.     Dividing  each  by  19  we  have 

their  mean  values  corresponding  to  one  foot  in  length  of 

scarp.     Let 

N  M  F  F'  „. 

— =129.35  =  ?i;  —  =  499.91  =w;  —  =  16.274  =  /^;  —  =  44.765  =  7^  ; 
19  '19  19  19 

the  known  height  of  scarp  from  (5'.50)  to  (43')  =  37'.5=Z'  ; 
the  known  thickness  of  scarp=8'=(?'. 


2  84  THEORY     OF    THE    ARCH. 

,r=tl»o  distance  between  the  exterior  face  of  the  scarp  and 
the  i)oint  where  tlie  curve  of  pressure  cuts  the  1)ase. 
Let  us  first  sujipose  the  pier  2'  by  4',  which  supports 
one  half  the  weiglit  of  tlie  coniniunication  arches  and  their 
res])ective  loads,  to  form  an  integral  part  of  the  scarp. 
AVe  have,  art.  64, 

,^^M£_-H^+.'«^Zt_^=3'.o56.  (44) 

lie-\-n 

This  is  the  equation  of  the  curve  of  pressure  in  the 
pier  (scarp),  in  Avhich  ^,  v^m,  F^  and  F'  are  constant,  and 
//,  ?,  and  /'  vary  by  equal  diflPerences. 

Giving  to  //,  /,  and  Z',  the  values  which  corres])ond  to 
the  bottom  of  the  foundations,  viz.,  /<=48  ;  /=22'.13  ;  I  = 
36'.13,  wefinda?=2'.13. 

As  the  foundations  extend  4'  in  front  of  the  scarp,  we 
see  that  the  curve  of  pressure  passes  very  nearly  through 
the  middle  of  the  lowest  course  of  masonry,  its  best  possi- 
ble situation.  Consequently,  the  scarp  is  in  no  danger  of 
rotary  motion. 

Let  us  now  suppose  the  piers  2'  X  4'  to  T)e  entirely  sep- 
arate from  the  scarp,  as  in  fact  they  are.     We  have 


a?=: 


he 

Giving,  when  7/  =  3r.50  ;  /=11'.G3  ;  /'=2;r.G3,  .t=1'.21 
"  7i=48'.00;  Z=22'13;  Z'=36'.13,  xj^O'.ir^ 
"      A=:32'.00;  /=C'.13;    r=:20'.13,        a'=2'.04 

The  lower  portions  of  both  of  these  curves  are  sketched 
on  fig.  18  ;  the  outer  curve,  c  c',  corresponding  with  these 
last  results,  the  inner  curve,  c  c,  corresponding  to  the  first 
supposition.  The  distance  of  this  cui-ve  from  the  surface, 
at  the  point  wliere  it  approaches  the  surface  most  nearly, 
is  the  best  measure  of  the  stability  of  the  sustaining  a\  all. 
At  t'  the  distance  c'  t'  is  l'.21=t<8')  X  0.151.  Tliere  is  no 
danger  of  the  i)ier  or  scarj)  overturning;  but  there  are  two 
other  points  to  which  we  must  direct  our  attention. 


STABILITY     OF    SCAlirS.— EXAMPLE.  '285 

(I).  The  liorizontal  joint  t  t',  reference  (5'.50),  may 
open  on  the  inside  and  allow  the  scarp  to  move  laterally 
throngli  a  certain  angle  around  c/,  near  /',  as  center. 

(II).  The  bricks  at  t'^  tlie  part  most  compressed,  may 
be  crnslied  by  the  superincumbent  weight. 

As  to  the  first,  we  can  make  no  estimate  of  the  extent 
of  angular  motion,  not  knowing  the  rate  of  compression  of 
Ijrick  and  concrete  masonry  under  a  given  pressure. 

As  to  tke  second,  the  entire  weight  suj^ported  by  the 
joint  t  t!  is,  /<6=300  cubic  feet  of  masonry. 

The  pressure  is  greatest  at  t! ;  it  is  0  at  the  distance 

3a.^=o'.63   from  t! ;    the  mean  pressure  is  lie  divided  l)y 

three  times  x ;    the  pressure,  per  unit  of  surface,  at  t\  is 

double  the  mean  pressure.     Calling  this  pressure  per  unit 

he 
of  surface  at  t\  p,  we  liaA^e  2)=2  X---=165.29=18,182. 

Odii 

pounds,  supposing  one  cubic  foot  of  the  mixed  masonry  to 
weigh  110  pounds. 

This  pressure,  about  126  pounds  per  square  inch,  is 
rather  too  great,  l)ut  probably  does  not  exceed  the»allowed 
limit,  one  tenth  the  crushing  force. 

There  are,  however,  some  elements  of  stability  ^^hich 
we  have  not  taken  into  consideration. 

We  are  warned  by  the  cracks  often  seen  in  old  works, 
not  to  rely,  in  any  degree,  upon  adhesion  of  mortar  in  the 
communication  and  casemate  arches.  But  there  is  another 
force  which  can  hardly  fail ;  viz.,  adhesion  in  the  joints  of 
the  scarp.     This  force  is 

|c'X6-=iX25x(8/=2GG.67, 
and   the    value  of    x  becomes,  at   the  joint   t  t\   where 
7^=:3r.50, 

he 
which  reduces  the  pressure,  per  square  inch,  at  /',  to  72.75 
pounds.     To  this  last  pressure  we  ought  to  add  the  reac- 
tion of  adhesion. 


23(3  THEORY     OF    THE    ARCH. 

PRESSURE    UPOX    THE    OUTER    PIERS    OF   THE    COMMUXICATIOX 

ARCHES. 

These  piers  Lave  an  area  in  tlie  liorizontal  section  of 
2'X4'  =  8,  and  each  one  of  them  has  to  sustain  the  whole 
vahie  of  iV  computed  al)ove. 

They  are  subject,  therefore,  to  a  pressure  of  30 7. '21  per 
s(iuare  foot ;  that  is,  each  square  foot  bears  a  weight  equal 
to  that  of  a  column  of  the  same  material  one  foot  square 
and  307 '.21  high  ;  a  pressure  of  33,793  pounds  per  square 
foot,  or  234.67  pounds  per  square  inch. 

SEGMENTAL    ARCHES APPROXIMATE    FORMULAE. 

85.  Tables  ^and  E'  give,  in  most  cases  with  sufficient 
accuracy,  either  directly  or  Ijy  proportional  parts,  the 
thrust  of  the  segmental  ring  of  equal  thickness  through- 
out, table  E,  and  of  the  same  I'ing  loaded  in  masonry  up 
to  the  level  of  the  extrados  at  the  crown,  talde  E'. 

But  j:hose  tables  have  not  been  extended  to  very  flat 
arches,  the  last  column  in  l)oth*  corresponding  to  -v=16/', 
and  ^'  =  14°  15';  nor  do  they  apply  very  well  to  cases  in 
which  -s^  exceeds  10/,  or  v  is  less  than  22°  37'  10". 

"When  it  Ijecomes  necessary  to  make  an  independent 
calculation,  and  to  ascertain  the  thrust  without  the  aid  of 
these  tables,  the  exact  methods  already  given  are,  it  must 
be  confessed,  rather  complex  and  tedious. 

We  shall  now  give  a  much  shorter  method,  sufficiently 
exact  for  all  those  cases  in  which  v\  the  semi-angle  at  the 
center,  does  not  exceed  30°;  and  applicable,  witli  little 
error,  t(j  much  larger  values  of  'v\  ])articularly  wlicn  the 
span  is  small,  as  it  usually  is  in  furtifications. 

The  cii-cidar  arc  m  ^,  fig.  15,  departs  Imt  little  from 
the  parabola  having  its  vertex  at  Z*,  and  passing  through 
the  point  in.  The  equation  of  momentsi  which  determines 
the  thrust,  is — 


b 


SEGMENTAL     ARCHES.— APPKOXIMATE    FORMULAE.  287 

FY.a  wi-'=moment  a  m  m  n  r  <^r''— moment  m  h  lu  m  ; 
Qn  beino;  tlie  center  of  moments.  Now  the  moment  of  the 
parabolic  surface  m  h  ai  m^  in  relation  to  rn^  is  |  'i/i  in  X 
h  m'x|Xw;'M'  =  jy-6y,  (6"= 2  mm^  ;  and  the  parabolic  sur- 
face m  h  m  m^  is  -f  m  m  X  iii  h^=^\-sf. 

As  the  parabola  is  wholly  below  the  circular  arc  at  all 
points  between  h  and  m,  we  in  effect  suppose  the  arch  to 
be  a  little  heavier  than  it  really  is  ;  and  shall  neutralize 
the  error  in  part,  or  more  than  neutralize  it,  by  adding  the 
small  triangle  m  n  t^  to  that  part  of  the  arch  which  is  on 
the  left  of  the  center  of  rotation. 

Let  airL=^y ,'  ^'= radius  of  the  intrados;  ^= radius  of 

the   extrados;    A^= — ;  if  the  arch  increase  in  thickness^'  ^-^-^ 

towards  the   pier  or  springing  line,  -A"r=:-,  d  being  thejk 

thickness  of  the  arch  at  the  crown;  'y'=the  semi-angle 
at  the  center.  We  have,  in  relation  to  m  as  center, 
moment  am!  mtnr  a=^yxlt  sin.  v\r  sin.  v'—^H  sin.i'')  = 
iyXlfXs\l-iJQ=iy.^XX-iX'').  We  have,  there- 
fore, as  the  thrust  of  the  segmental  arch  loaded  u}>  to  the 
level  of  the  extrados  at  the  crown. 

Illustrations.  Suppose  .5'=10/;  giving  r=l?if ;  v'  — 
22°  37'  10";  and  let  X=1.10. 

The  above  formula  gives,  i^^^-^X 0.07845  =  ;- x 0.04043 
Table  E'  gives  for  the  same  case,    .     .    i'^=r-x  0.04655 

Error,  =/'^XO.OU013 

Suppose  5=G/;  r=5/;  ^'=36°  52' 10";  A^=1.10. 
The  above  equation  gives,     .         .         F=z i^  x  0.07820 
Table  E'  gives  for  the  same  case,      .     F—i^ X  0.07724 

Error,         =/-'x  0.00096 


*  The  point  r,  omitted  on  the  left  side  of  fig.  15,  is  vertieally  over  n  on  the  hor- 
izontal through  a. 


288  THEORY     OF    TUE     ARCH. 

TliU  approximate  formula,  (47),  gives  with  all  tlesirable 
accuracy  the  thrust  of  the  caseuiate  arches  of  Fort  Jeffer- 
son, art.  78. 

SO.  The  thrust  of  the  segmental  arch  loaded  hori- 
zontally np  to  a  plane  passing  at  the  distance  t  above  the 
crown  of  the  arch,  is 

F=\.r{K-\IC-^'^l)+^6\E-^Ji:^)^         (48) 

In  all  cases  y=.f-{-J. 

87.  The  thrust  of  the  segmental  arch  loaded  up  to  any 
plane  D'  M',  fig.  19. 

Let  vi  R'—E;  m  a=li' ;  n  r—E";  the  known  dis- 
tance ///  i=.m  )i  Xsin.  v^=^D.     We  have 

The  last  term  is  usually  small,  and  in  very  light  arches 
may  be  omitted  altogether.  D  and  E"  can  always  be 
taken  with  sufficient  accuracy  from  a  drawing  of  the  arch. 
Equation  (49)  of  course  includes  (48),  but  is  more  general 
in  its  charactei*.  It  does  not  contain  the  ratio  /iT,  and  is 
not  founded  upon  the  supposition  that  the  arch  is  of  equal 
thickness  throughout.  Moreover,  it  is  strictly  accurate  as 
to  the  moment  of  that  part  of  the  arch  and  its  load  which 
overlies  the  skewback ;  the  little  triangle  ntrti  no  longer 
forming  any  pai-t  of  this  moment. 

Api)lied  to  the  case  in  which  the  surcharges  is  hori- 
zontal, we  have  E=E;  E"=E—mnXco^.v';  in  all 
citses  y=f-\-d=z((m'. 

Illustration.  Let  us  apply  (49)  to  the  upper  casemate 
arch  of  Fort  JeftV-rson  (see  arts.  81,  82). 

The  tlirust  given  by  (49)  is      .      7^  =  2G.S1 
The  exact  thrust,  art.  81,  is  .       .  7^  =  20.57 

The  diftVronce,         =  0  24,  always 


( 


i 


SEGMENTAL    ARCHES.— ArPROXIM  ATE    FORMULAE.  289 

in  fiivor  of  stability,  is,  we  see,  very  small,  notwith- 
standing tlie  great  extent  of  the  semi-angle  at  the  center, 
^'=43°  36'  10". 

We  are  therefore  dis]')ose(I  to  recommend  formula 
(49)  for  exclusive  use  in  calculating  the  thrust  of  the  seg- 
mental arches  of  fortifications,  when  the  thrust  can  not  he 
obtained  from  the  tables  of  circular  or  other  arches  con- 
tained in  this  2^^^per. 

88.  The  sliding  thrust  of  segmental  arches,  when  the 
angle  of  rupture  extends  to  the  springing  line,  is,  using  the 
notation  of  the  preceding  article, 

P=(is{£J+i;'-^f)  +  ~{£;'+i:"))xcotmg.{i>'  +  30°)         (50) 

We  can  generally  tell  in  advance,  whether  the  true 
thrust  is  due  to.  rotation  or  sliding ;  if  not,  it  will  be  neces- 
sary to  calculate  both. 


THICKNESS'  OF    PIER — APPROXIMATE   FORMFLJE. 

89.   For   notation,  see   art.   82.      Still   reo^ardino;  the 
intrados  m  Z*,  fig.  19,  as  a  parabola,  we  have 

«=K-E'+-^'-i/')  (51) 

m=-,'j6\i;+iE'-i/)  (52) 

6'  is  in  all  cases  the  span,/  the  rise. 

Applying  formulae  (49),  (51),  (52)  to  the  upper  case- 
mate arch  of  Fort  Jefferson, 

We  find,  when  ^  =  1,  equation  (31),         .         (?=3'.59 

The  exact  formulae,  art.  82,  gave  us,    .         .     t^=3'.58 

When  ^  =  2,  equation  (31).  .  .         <?=G'.T4 

The  exact  formulae,  art.  82,  gave         .         .     e={j'.7- 

It  thus  appears  that  while  the  approximate  methods 

require  far  less  lal)or  than  the  exact,  they  lead  to  almost 

identical  results.     The  error  committed  in  obtaining  the 


290  TIIEOR\'     OF    THE    ARCH. 

tlini^t,  i>'  Ixilaiiced  in  part,  or  more  than  balanced,  wlien 
^ve  ai)ply  (51),  (52)  to  tlie  determination  of  the  tliiekness 
of  pier.*  Equations  (51),  (52)  may  be  used  when  the 
tlirust  has  been  obtained  by  exact  methods ;  but  the  error 
in  tlie  thickness  of  pier  will  be  somewhat  increased. 

The  formuliTi  given  for  the  thickness  of  pier  in  art.  64 
are,  as  we  have  repeatedly  stated,  universal. 

The  principle  of  these  approximate  methods  has  already 
been  applied  in  calculating  the  stability  of  the  scarp  wall 
of  Fort  Jefferson ;  and  we  have  giMie  over  the  same  ground 
l)y  the  more  laborious,  exact  modes  of  computation.  The 
results  were  almost  identical ;  but  we  could  not  always  look 
for  such  close  approximation. 


SECTION  V. 

ELLIPTICAL    ARCHES. 

90.  Elliptical  arches  are  but  little  used  on  fortifications, 
where  economy  and  stability  are  more  regarded  than 
architectural  effect.  They  are,  however,  sometimes  used 
in  stone  and  brick  bridges  on  great  thoroughfares,  and 
particularly  in  the  neighborhood  of  large  cities. 

The  rise  of  the  arch  of  almost  all  long  bridges,  is  less 
than  the  half-span. 

There  are  three  principal  varieties  of  the  intrados  : 

1st.  The  ellipse ; 

2d.  The  segment  of  a  circle ; 

3d.  The  3,  5,  7,  ttc,  centered  arch. 

The  2d,  or  segmental  arch,  is  the  strongest,  the  most 
economical,  and,  in  general,  the  best.  It  is  more  easily 
})uilt,  less  liable  to  change  its  form  after  the  removal  of 
the  center,  on  receiving  its  final  load,  or  any  variable  and 
occasional  load,  and  has  less  horizontal  thrust  at  the  key 


ELLIPTICAL     ARCHES.  291 

than  any  other  arch  of  the  same  span  and  rise  ;  and  its 
appearance,  when  the  rise  is  small  in  relation  to  the  span, 
is  more  agreeable  to  the  eye  than  tliat  of  the  flat  ellipse. 
This  variety  has  already  been  disposed  of  We  will  liere 
add  the  remark,  that  the  segmental  arch  or  ring  should 
increase  in  thickness,  from  the  key  to  the  springing  line,  at 
snch  a  rate  as  to  become,  if  continued  to  60°  from  the  key, 
about  one  half  greater  than  it  is  at  the  key.  Such  increase 
will,  in  general,  not  only  insure  the  requisite  stiffness  or 
stability  of  form  in  the  arch  itself,  but  will  nearly  equalize 
the  pressure,  per  unit  of  surface,  upon  the  joints  of  the 
key  and  springing  line. 

Should  the  intrados  of  the  segmental  arch  extend  more 
than  60°  from  the  key,  the  augmentation  of  thickness  must 
continue. 

Arches  of  the  3d  class  are  all  appi'oximations,  more  or 
less  close,  to  the  ellipse  having  the  same  rise  and  span. 
With  five  or  more  centers,  the  approximation  becomes 
almost  an  identity;  and  we  may  regard  the  ellipse  as 
representing  all  these  arches,  that  is,  as  having  the  same 
thrust  and  requiring  the  same  thickness  at  the  key  and  the 
same  thickness  of  pier. 


THRUST    OF   THE    ELLIPTICAL    AECH    WITHOUT   LOAD. 

Figure  20. 

91.  Let  A  C^  the  half  span  and  semi-transverse  axis, 
=:r  •  C  h^  the  rise  and  semi-conjugate  axis,==//  a  h^  the 
thickness  at  thekey,=:f/.  Let  us  compare  this  arch  with  a 
circular  arch  of  the  same  span,  and  of  a  thickness,  u'h'—-d\ 
at  the  key,  as  much  greater  than  the  corresponding  thick- 
ness of  the  elliptical  ring,  as  the  half-span  is  greater  than 
the  rise. 

And  let  us  further  suppose  the  vertical  depth  of  the 
elliptical  ring  to  bear  a  constant  ratio  to  the  depth  of  the 


0()'2  TUEORY    OF    THE    ARCH. 

auxiliary  circular  arch,  both  measured  on  tlie  same  vertical 
line  ;  so  that,  on  any  vertical  line  m  ;•',  Ave  shall  have, 

m  r  :  7)1  r'  : :  ah  :  a'  h'  :  :/* :  r. 

This  re(|uires  the  extrados  of  the  elliptical  arch  to  be 
another  ellipse,  havhig,  for  its  axes,  Ca=f-\-d,  and  CB= 

Ca'  —  r-\—.y.(L  This  arrangement  gives  a  continual  aug- 
mentation, not  too  great,  to  the  thickness  of  the  elliptical 
arch.  Comparing  together  the  segments  m  r  a  h,  m!  r  a  h\ 
included  between  the  vertical  of  the  key  and  any  other 
vertical  m  r\  we  see  that  their  horizontal  dimensions  are 
the  same,  while  their  vertical  dimensions  bear  the  constant 
ratio  of  f  to  r.  Consequently,  the  surfaces  of  those  seg- 
ments sustain  the  same  constant  ratio,  and  their  centers  of 
gravity  are  in  the  same  vertical  line.  Projecting  m  and 
,  7)1  horizontally  on  C  a  at  t  and  t\  and  disregarding,  for 
the  present,  the  negative  influence  upon  the  thrust  of  the 
surfaces,  nearly  triangular,  711  n  r  and  in  n  7-  ;  designating 
by  aS',  the  surface  m  r  a  h;  by  ^;,  the  distance  of  the  center 
of  gravity  of  this  surface  from  the  vertical  through  in;  by 
y,  the  lever  arm  a  t;  by  S\  p\  y'  the  corresponding  surface, 
distance,  and  lever-arm,  a  t\  of  the  circular  arch, — we  have 
for  the  thrust,  F^  of  the  elliptical  arch,  corresponding  to 
any  position  of  the  vertical  line  711  r', 

F=—,  and.  for  the  circular  arch,  F'=i—^r- 

f 

But  we  have  already  found  2>=p\  S=--S\  and  we  have 

/:r:  C h  :  CV  '.:  C a '.  C a  ::  C t  :  C t'  :  Ca-Ct{=y)'. 

C a'  —  Ct'  {—y).     Consequently,  //=■  -Xf/ ;  and  F—F'. 

This  relation  exists  for  all  positions  of  the  joint  of  rup- 
ture; hence,  the  maximum  or  true  thrust  of  the  two 
arches  Avill  be  the  same,  and  we  are  able  to  announce  this 
principle : — 


ELLIPTICAL    ARCHES.  293 

The  thrust  of  a  soiiicircular  arch,  of  equal  thklmess 
throiujlout,  and  tuithout  load,  is  nearly  equal  to  the  thrust 
of  an  elliptical  arch  of  the  same  sqxin,  and  of  a  vertical 
depth  at  the  hey  and  at  every  other  point  as  much  less  than 
the  depth  of  the  circidar  arch,  on  the  same  vertical  lines,  as 
tlie  rise  of  the  elliptical  arch  is  less  than  the  half  span. 

We  shall  therefore  be  able,  with  little  error,  to  ol)tain 
the  thrust  of  unloaded  elliptical  arches,  from  table  A. 

92.  The  result,  however,  thus  obtained,  i-equires  a 
slight  addition.  We  have  neglected  the  difference  in 
effect  of  the  small  surfaces  m  n  r,  m'  n!  r,  which  tend  to 
diminish  the  thrust  of  the  arches  to  which  they  respec- 
tively belong.  If  the  moments  of  these  two  surftices  in 
relation  to  the  vertical  through  m,  like  the  moments  of 
the  segments  m  r  a  h,  m'  r  a!  h',  stood  in  the  proportion  of 
/  to  i\  no  correction  would  be  necessary.  But  such  is  not 
the  case. 

Draw  the  tangents  m  o,  m!  o,  to  \\\^  intrados  of  the  ellip- 
tical and  of  the  auxiliary  circular  arch,  meeting  on  the 
transverse  axis  or  horizontal  of  the  springing  line  at  o;  draw 
the  normal  nip>  to  the  ellipse  intersecting  A  C  at  p  ;  pro- 
long the  line  p'  m  to  n,  the  extrados  of  the  elliptical  arch : 
and  the  line j^'  ni  to  n'  the  extrados  of  the  circular  arch. 
n  and  ti  are  evidently  on  the  same  vertical  line.  Let  v 
represent  the  angle  between  any  joint  m  n — sui)posed  to  be 
normal  to  the  intrados — and  a  vertical;  v'  the  corre- 
sponding angle  n  m  r ,  or  m!  C a,  of  the  circle ;  and  V the 
angle  n!'  711  r.    We  have  fr^anglep  0  m;  t''=angle  i)  om'/ 

f  f 

T^=  angle  jy  ni  p '^   tang.  'V=-_tang.i)'y  tang.  1^=- tang. •v^ 

^    tang. 'y'.     From  these  relations  v  and  F^  are  easilv  cal- 


L 


culated  when  v  is  given. 

The  triangles  m  n  r,  rn  n'  r' ,  have  the  same  altitudes, 
and  bases  m  r,  m!  r' ,  in  the  proportion  of  y  to  r.  Conse- 
quently, they  have  the  same  effect  upon  the  thrust ;  and 


29i 


THEORY    OF    THE    ARCH. 


the  required  correction  consists  in  adding  to  tlie  tbrust,  as 
given  l)y  table  A,  the  effect  of  the  triangle  in'  it  n'.     Let 
A  represent  the  required  addition.     We  have 
(/y?  11  y  sm/  V  —{in  it  y  sin.-  r 


6 


(53) 


^ow^  in  all  the  cases  likely  to  occur  in  practice,  the 
angle  of  rnjiture  corresponding  to  the  maximum  thrust,  is 
in. the  neighborhood  of  60°;  and  we  shall  calculate  the 
value  of  A  on  that  supposition,  ^representing  the  ratio 
of  the    two  radii  of  the   circular  arch,   we    have,   when 

v'=(^0%  m'r'=r(^/X^-0.n  -  0.50)  ;    a't'  =  r{K-  0.50)  ; 
m'n=r{K-l) ;  ;///^"=r(|'7r--sin.X60°-  r)-cos.(60° 

-F));tang.  F-^^' 

We  subjoin  the  values  of^  corresponding  to /=:!;■, 
and  to  all  values  of  ^between  1  and  1.60. 


Table  A'. 


A 


Values  of  — -  to  be  added  to  the  Coefficients  of  r-  givex  nv  the 
r' 

4Tn  Column*  of  Table  A,  calculated  on  the   supposition   that 


/_ 


h 


,  Value  of 

-*5 

Value  of  ^ 

--; 

A 

.=,.;, 

^- 

A=l.-. 

A 

1.01 

0.00000 

1.16 

0.00092 

1.31 

0.00510 

{1.46 

0.01343 

1.02 

0.00000 

1.17 

0.00108 

1.32 

0.00552 

1.47 

0.01414 

i.o;j 

0.00000 

1.18 

0.00126 

!   1.33 

0.00596 

1.48 

0.01487 

1.04 

0.00002 

1.19 

0.00144 

1.34 

0.00642 

1.49 

0.01562 

1.05 

0 . 00004 

1.20 

0.00165 

1 .  35 

0.00690 

1.50 

0.01639 

1.00 

O.OOOOG 

1.21 

0.00188 

1.36 

0.00740, 

1.51 

0.01718 

1.07 

0.00009 

1.22 

0,00212 

1.37 

0.00792 

1.52 

0.01799 

1.08 

0.00014 

1.23 

0.00238 

1.38 

0.00845 

1.53 

0.01882 

1.09 

0.00019 

1.24 

0.00266 

1.39 

0.00900 i 

1.54 

0.01967 

1.10 

0.00025 

1.25 

0.00295 

1.40 

0.00957 

1.55 

0.02054 

1.11 

0.0003.3 

1.26 

0.00326 

1.41 

0.01016 j 

1.56 

0.02143 

1.12 

0.00042 

1.27 

0.00358 

1.42 

0.01077  1 

1.57 

0.02234 

1.13 

0.00052 

1.28 

0.00393 

1.43 

0.01140 

1.58 

0.02328 

1.14 

0.00064 

1.29 

0.00430 

1.44 

0.01206 

1.59 

0.02423 

1.15 

0.00078 

1.30 

0.00469 

1.45 

0.01274 

1.60 

0.02519 

ELLIPTICAL     ARCHED.  205 

The  value,  A\  of  A  corresponding  to  any  other  rela- 
tion of  /  to  r,  will  be  given,  with  sufficient  accuracy,  by 
the  followinor  formula : 

.4'=:2.1(1-'-). 

This  last  formula  will  give  thrusts  slightly  in  excess 

for  values  of--  between  1  and  ^,  and  thrusts  a  little  too 
r 

f 
small  for  values  of--  less  than  i. 
r 

93.  Reca2:)itulation.  To  find  the  thrust  of  the  unloaded 
elliptical  arch,  the  extrados  being  an  ellipse  similar  to  the 
intrados ;  ?'=tlie  half  span;  /=the  rise;  <:/=the  thick- 
ness at  the  key : 

Look  in  table  A,  4th  column,  for  the  coefficient,  C,  of 

^■^,  opposite  ir=14-- ;  to  this,  add  the  product  C'  =  '2(l  — 

i-^  X^-,  ^  being  taken  from  the  above  talde  A'  opposite 

the  same  value  of  K.     Then  F^r\C^ C). 

Example.  r=10' ; /=6' 8"  ;  rt^=l'.50;  hence  X=l  + 

^=1.225. 

Value  of  (7,  4th  column  of  table  A,  mean  be- 
tween 7i"=il.22  and  7t^=1.23     =0.12044 

Value  of  6"  =  2(1 -I)  X  0.00225;  4"  ^^^^"o  ^^^^ 

mean  between  A"=1.22  and  1.23,  table  A'     =0.00150 

Total  thrust  i^=:/--X  0.12194 
M.  Audoy  gives  as  the  thrust  of  a  three-    , 
center  arcli  of  the  same  span  and  rise 
and  thickness,  the  intrados  being  de- 
scribed with  three  arcs  of  60°  each,       F^r'' X0.125G9 

Difference,      =:r-X  0.00375 
=r  about  three  per  cent,  of  the  true  thrust. 


29(3  THEORY     OF    THE     ARC  EI. 

Example  2.    /^lO';  f=-)\  J='2\  lioiice   /r=14-|= 
1.40. 
Value  of  C\  4tli  column  of  talde  A,  opposite 

7l'=1.40     =0.16167 
Value  nf  r  =2(1 -;L)x 0.00957  tal.le  A' op- 
posite 7^=1.40     =0.00057 

i^z=/-X0.17124 
M.  Au(l(n'  dves  as  the  thrust  of  a  five-center 
arch  of  the  same  span,  rise,  and  thickness, 

i^=/--X  0.17914 


Difference =i^=/-  X  0.00790 
=  al><>ut  4.V  per  cent,  of  the  true  thrust. 

The  rule  stated  above  gives  immediately,  and  with  all 
desirable  accuracy,  the  thrust  of  elliptical  arches,  unloaded. 
Table  H  contains,  all  calculated,  the  rotation  thrusts  in 
two  systems  of  ellijitical  arches,  corresponding  respectively 

to'-=i  and-    =?.     The  first  colunni  gives  the  quotient 

r      '  r 

of  the  s])an  divided  by  the  thickness  at  the  key,  this 
(piotient  being  the  proper  measure  of  the  lightness  of  the 
arch.     That  table  seems  to  re(j[uire  no  ex])lanatiou. 

SLIDING    THRUST   OF    UNLOADED    ELLIPTICAL    ARCHES. 

Figure  20. 

94.  It  appears  from  ta])le  A,  that  the  rotation  thi'ust 
of  the  unloaded  circular  arch  exceeds  the  sliding  thrust  for 
all  values  of  7^  less  tlian  1.45.  It  appears  from  table  A', 
art.  92,  that  the  rotation  thrust  of  tlie  elli])tical  ring, 
bounded  by  similar  ellipses,  is  greater  than  the  rotation 
thrust  of  the  circular  arch  resting  upon  the  same  sjian  and 
having  a  thickness  at  the  key  as  much  greatei-  than  the 
tliickness  of  the  ellijitical  arch  as  the  half-span  is  greater 
than  the  rise ;  and  that  the  difference  increases  as  K 
increases. 


ELLIPTICAL    ARCHES.  OQ'j- 

Witliont  stopping  to  demonstrate  it,  we  here  state  the 
fact  tliat  tlie  sliding  thrust  of  the  elli]-)tical  arch  is  always 
less  than  that  of  the  auxiliary  circular  arch  above  de- 
scribed. Putting  these  facts  together,  we  can  give  the 
following  rule  as  perfectly  safe,  though  liable  to  give  a 
thrust  too  great : 

Find  the  rotation  thrust,  art.  93.  Should  this  thrust 
be  less  than  the  sliding  thrust  found  in  table  A,  opposite 

7 

J^=  1  +-?,  adopt  the  latter  as  the  true  thrust. 


THICKNESS    OF   PIEK ELLIPTICAL    AECIIES UNLOADED. 

Figure  20. 

95.  Let  i^=the  thrust ;  //=:the  height  of  pier  from  the 
l^ase  to  the  springing  line ;  l=.t\\Q  lever-arm  of  the  thrust 
or  elevation  of  a  above  the  base  of  the  pier;  7i=the  sur- 
face of  the  semi-arch  ^  ^  «  Z* ;  m=the  moment  of  that 
surface  in  relation  to  A  ;  ^=:the  coefficient  of  stability; 
6= the  unknown  thickness  of  j)ier. 

7  7  7^ 

7i=|,,7(2+^);    m=rV/(.5708-.2146x^-JX^^);     (54) 
lM+7ie-^m=hFl 


THEUST  OF  ELLIPTICAL  AECIIES  SUSTAINING  A  LOAD  OF 
MASONRY,  OR  OF  EQUAL  WEIGHT  WITH  MASONRY,  RISING 
ON  EACH  SIDE  OF  THE  CENTRAL  RIDGE,  TO  A  ROOF 
TANGENT   TO    THE   EXTRADOS.       FigiiVe    21. 

96.  Let  us  compare  the  given  arch  with  a  circular  arch 
of  the  same  span,  and  of  a  thickness  at  the  key  as  much 
greater  than  the  thickness  of  the  elliptical  arch  as  the 
half-sj^an  is  greater  than  the  rise ;  and  let  us  suppose  the 


298  THEORY     OF    THE    ARCH. 

loads  of  tlie  two  arc-lies  to  sustain  this  same  relation  in  tlieir 
vertical  deptlis.  We  suppose  tLe  tliickness  of  the  two 
arches  at  the  springing  line  to  be  the  same,  which  requires 
the  extrados  of  the  ellii)tical  arch  to  l)e  an  ellipse  similar  to 
the  intrados.  R  P,  the  roof  of  the  given  arch  intersect- 
ing the  horizontal  of  the  springing  line  at  P  ;  draw  P  B! 
tangent  to  the  extrados  of  th^  circular  arch.  The  two 
arches  thus  constructed,  sustain  to  each  other  the  relation 
described  above.  We  have,  by  supposition,  a  h  :  a  V  : : 
f :  r.  Draw  the  vertical  d  V  passing  through  the  points 
of  tangency  D  and  D' . 

Draw  any  other  vertical  line,  ^;  n  cutting  at  m  the 
intrados,  at  r  the  extrados,  and  at  ?<  the  roof  of  the  ellip- 
tical arch  ;  at  7?i',  r',  ii  the  corresponding  parts  of  the 
circular  arch.     We  have 

2^11  :2^i('  •.■.ilD-.dlJ'  :'.2^r'.pr'  '.'  C  a  :  Caw  Ch\  CV 
'.'.f'.  V  : '-  X)  m  \  iJ  m .     Hence  ni  u  :  ni  u'  : :/' :  r. 

Moreover  projecting  in  and  m'  horizontally  on  the  ver- 
tical through  C\  at  t  and  t\  we  have,  as  in  art.  91,  a  t  \ 

Pursuing  precisely  the  same  course  of  reasoning  as  in 
art.  91,  we  see  that  the  thrusts  due  to  the  segments  m  u 
P  ^,  m  111  P'  y  are  the  same  in  the  two  arches,  wherever 
the  vei-tical  m  tyi  be  drawn.  Consequently,  the  maximum 
thrust  of  the  two  arches  is  the  same,  and  we  can  announce 
this  principle : 

The  ilirust  of  an  elliptical  ai-cli  loadtd  in  masonry  up 
to  a  plane  tangent  to  the  extrados^  supposed  to  he  similar 
to  the  intrados^  is  nearly  equal  to  the  thrust  of  a  circular 
arch  of  the  same  span^  and  of  a  thicMess  of  ring  and  of 
load  at  the  hey  as  vniclt  greater  than  the  corre-^ponding 
depths  of  the  elliptical  arch^  as  the  half  span  is  greater  than 
the  rise;  the  load  of  the  circular  arch  also  rising  to  a  plane 
tangent  to  its  extrados. 

We  can,  therefore,  in  most  cases,  obtain  the  required 


ELLIPTICAL    ARCHES.— LOADED.  299 

tlinist  from  table  F.  The  re^^ult,  however,  thus  obtained, 
will  require  a  slight  addition,  viz : 

moment  on  m'  of  the  surface  m  n'  i  i"  n'  m        .       /-_n 

■^~, -z^A.     (55) 

lever  arm  at 

which  may  he  calculated  on  the  supposition  that  tlie  angle 
of  rupture  of  the  circular  arch,  is  60°. 

It  will  be  best  to  construct  the  diagram  and  obtain  the 
elements  of  this  calculation  by  protraction. 

For  obvious  reasons  the  addition  required  will  gene- 
rally exceed  but  little  the  values  given  in  table  A',  art.  92. 

Rule. — Suppose  the  angle  C R  P^^T  to  be  given  by  its 
tangent.      Let  angle  0  H'  P=^I;  half-span =?■;  rise=/; 

thickness  at  the  key=a^=c/.    We  have  taug^.  7=— tano:.  /', 

which  gives  I. 

01:)tain  from  table  F  the  coefficient  0  of  f^  correspond- 

iuo'  to  this  value  of  Zand  to  Iv=i1-\-—t\  to  this  add  the 

O  T 

(       f\    A  A 

product  21  1  — ^  I  X-j^:^,  -y  being  taken  from  table  A', 

art  92,  opposite  the  same  value  of  K. 

Then  the  thrust  F^r\O^Cy 

To  this  we  ought  to  add,  when  n  i\  n"  i'\  60°  from 
the  key,  have  any  considerable  magnitude,  the  effect  upon 
the  thrust  due  to  the  trapezoid  n'  i'  i"  n". 

97.  Example:   r=\^  feet;/=6|  feet;  7'=i60° ;  d= 

1'.30;  X=^l+^^=1.195;  /=49°  6'  24"=49°.10f ;    50°- 

7=0°.89J. 

Coefficient  of  ;-^  table  F,for7i"=  1.195, 7=45°=  (7=0.25676 
"  "  "  "         7=50°      "     0.20942 


Difference,  0.04734 


300  THEORY    OF    THE    ARCH. 

5°:  O^.SOJ-::  0.04734  :  x  =0.00846 

Add  C,  as  above,  for  /r=1.195  ;  7=50°,  0.20942 

Ada  from  taUe  A',  art  92,  2(1-3)  X 0.00155,        0.00103 

Total  thrust,  i^=r-X  0.21891 

H.  Audoy  gives  as  the  tlirust  of  a  three-center 

arch  of  the  same  rise,  span,  and  h:>ad,  r"^  X  0.22075 

Diftereuoe  =  |  of  1  per  cent,     .         .         .      ^r^X  0.00184 

The  rule  given  above  can  only  be  used  when  tlie  angle 
/is  o-reater  than  45°,  or  but  little  less  than  that  limit. 

In  other  cases  it  will  be  best  to  investigate  the  thrust 
geometrically. 

The  effect  of  a  surcharge  of  uniform  vertical  depth 
may  be  obtained  from  the  talkie  in  art.  104 ;  but  the  addi- 
tion thus  found  will  be  a  little  too  large. 

Let  t  represent  the  depth  of  the  surcharge.     Look  in 

that  table  opjwsite  K=.\-\ — r  for  the  value  of  C;  the 
required  addition  will  be  A  —  rX—:C.   ' 


THICKNESS    OF   PIER.       FujiLVe    21. 

98.  The  general  formuhT3  of  art.  64  are  all  applicable 
to  the  elliptical  arch.  E^  E\  /,  tfec,  must  be  taken  from 
the  given  elliptical  arch.  The  values  of  n  and  m  will  be 
as  follows : 

m  =  /'Xi^-j-i/"j-//-^X 0.452065  I  ^'    ^ 


ELLIPTICAL    ARCHES.— SURCHARGED    HORIZONTALLY.       301 

ELLIPTICAL     ARCHES     LOADED    IIORIZONTALLY    UP     TO     THE 
LEVEL    OF   THE   EXTRAD03   AT   THE    KEY.       Fujurt  22. 

90.  This  is  tlie  most  eomnion  form  of  the  elli])tical 
arch,  ahnost  the  only  form  in  practical  use. 

Let  r^=^A  6'=the  half-si)aii  and  semi-transveise  axis 
of  the  ellipse  ; /=C'/>= the  rise,  and  semi-conjugate  axis; 
d—ah=\\\Q  thickness  at  the  key.  This  thickness  we  sup- 
pose to  be  constant.  The  calculated  thrust  will  be  a  little 
larg-er  than  it  would  have  been  had  the  extrados  been 
another  ellipse  similar  to  the  intrados. 

Let  us  compare  the  given  arch  with  a  circular  arch, 
surcharged  in  like  manner  horizontally,  having  the  same 
span,  and  a  thickness  at  the  key  as  much  greater  than  the 
thickness  of  the  elliptical  arch  as  the  half-span  is  greater 
than  the  rise. 

All  the  vertical  dimensions  of  the  auxiliary  circular 
arch  l)ear  a  constant  proportion  to  the  corresponding 
dimensions  of  the  elliptical  arch ;  so  that,  drawing  any 
vertical,  p  mi  in  ?/',  we  have 

/      aJ>  _  Cl>  _  Ca  _  1711  _  pm  _  irm^ 
'r~'clb'~  Ch'~~  Co! "'p it  ~pni  ~ m'u 

Consequently  the  surfaces  m  n  a  b^  ml  u'  a'  b\  are  also 
in  the  proportion  of/  to  r  ;  their  centers  of  gravity  are  on 
the  same  vertical  line  ;  their  lever  arms,  a  t^  a'  t\  or  m  ii, 
111  u\  are  in  the  proportion  of /to  r. 

These  surfaces,  therefore,  have  the  same  thrust  wherever 
the  vertical  line  be  drawn.  Tlieir  maximum  thrusts  will 
be  tke  same  ;  and  we  come  to  this  conclusion : 

Tlietlirust  of  an  elliptical  arcli  sustaining  a  load  of 
masonry,  or  of  equal  weight  tvitli  masonry,  rising  to  the 
horizontal  line  tangent  to  the  extrados  at  the  ley,  is  nearly 
equal  to  the  thrust  of  the  semicircular  arcli,  loaded  in  lihe 
manner,  having  the  same  span  and  a  thichiess  at  the  Tcey 
as  much  greater  than  the  thichiess  of  the  elliptical  arch  as 
the  half-span  is  greater  than  the  rise. 


302  THEORY     OF    THE    ARCH. 

AVl*  sliuU  tlierefore  be  able  to  obtain  from  t:il)le  D, 
with  littk'  labor,  the  thrust  of  elliptical  arches. 

The  thrust  thus  obtained  would  be  ])erfectly  coi-rect  if 
the  moments,  in  relation  to  the  vertical  through  ?><,  of  the 
two  surfaces  in  n  i  ?/,  m'  n'  i'  y\  stood  like  the  segments 
on  the  right  of  that  vertical,  in  the  relation  of/  to  r. 

But  this  is  not  the  case  ;  and  it  is  necessary  to  correct 
the  result.  In  calculating  table  D,  we  have,  in  eifect,  sub- 
ti-acted,  from  the  thrust  due  to  the  segment  r/i  u'  a  h\  the 
negative  influence  of  dl  n  i'  i(\  both  taken  at  the  angle  of 
maximum  thrust. 

For  our  ])resent  purpose  we  should  have  subtracted 
only  the  effect  oi  m'  ri'  i"  u\  which  stands  very  nearly  in 
the  rerpiired  relation  to  the  surface  m  n  i  u.  The  joint 
inn— ah— d  is  supposed  to  be  noi-mal  to  the  intrados  ; 
m'  n"  parallel  and  ecpial  to  m  n.  It  is  evidently  necessary 
to  ad(l  to  the  results  of  table  D  the  difference  of  the  effects 
of  the  two  surfaces,  m  n  i' u,  7)i  n"  i"  u' ;  or,  increjising 
this  difference  slightly,  the  effect  of  the  surface  s  s'  i'  ^",  of 
which  the  altitude,  m  u\  is  also  the  lever  arm  of  the  thrust. 
Kamino-  v  the  ans-le  between  ui  n  and  a  vertical,  v  the 

f 
angle  between  m  n    and  a  vertical,  we  have,  tang.  ^'=-7 

tang,  t;'/  and  for  the  thrust, 

i(f/ysin.^/-^/-in-r. 

in  which  6' is  taken  from  table  I),  and  v  is  supposed  to  be 
the  angle  of  maximum  thrust.  This  angle  never  differs 
much  from  00°.  We  are  at  lil)erty,  therefore,  in  all  cases, 
to  suppose  v'  =  60^;  Avhich  gives 

r- 


amrv  =1 ;  sm. 


^   2 


f 
1+3;;, 


ELLIPriCAL    ARCHES.— SURCHARGED     HORIZONTALLY.        303 

Tiil)le  G,  calculated  from  tlie  above  formula,  gives, 
either  directly  or  by  proportional  parts,  the  thrusts  of  all 
elli])tical  arches  in  practicable  use. 

The  first  column  is  the  quotient  of  the  span  dividetl  by 
the  thickness  of  the  arch  at  the  key,  this  quotient  being 
the  proper  measure  of  the  massiveness  or  lightness  of  the 
arch. 

100.  The  tal)le  on  the  following  page  gives  the  hori- 
zontal thrusts  of  elliptical  and  segmental  arches  of  the 
same  span,  rise,  and  thickness,  in  four  systems,  all  the 
arches  being  surcharged  horizontally. 

The  reader  will  perhaps  be  surj^rised  to  see  that  there 
is  but  little  difference  in  the  thrusts  of  the  two  kinds  of 
arches,  and  that,  in  very  light  arches,  the  difference  is  in 
favor  of  the  elliptical  intrados.  When  the  rise  is  one  fourth 
or  one  fifth  of  the  span  and  the  thickness  about  one  twenty- 
fifth  of  the  span,  the  thrust  is  nearly  the  same  in  elliptical 
and  segmental  arches. 

To  explain  briefly  the  manner  of  comparing  these 
thrusts  :  In  tables  E'  and  D,  r  is  the  radius  of  the  inti'a- 
dos  ;  in  the  following  table,  v  is  the  half-span.  Let  y"r=the 
half-span ;  /"^ithe  rise  ;  (^==the  thickness  at  key ;  ?''=the 
radius  of  the  intrados  of  the  segmental  arch  of  the  same 
span,  rise,  and  thickness  ;  A'=the  i-atio  of  the  two  radii  of 

the  se^rmental  arch  ;  K'—-^.     We  have 

F=r"  X  C=r-  X  \(A  +  y)  X  ^• 

From  this  last  formula  the  coeflicieuts  of  r^  in  the  fol- 
lowing table  have  been  computed  from  the  coefficients  of 
'?''^,  (r^),  in  tables  E'  and  D. 


304 


THEORY     OF    THE     ARCH. 


101.  lahle  of  horizontal  thrusts  of  eUiptlval  and  segmental 
arches  of  the  same  Sjxni^  rlse^  and  thicliiess  at  the  ley^ 
in  four  systems ;  surcharged  horizontal! ij  to  thehorizon- 
taJ plane  tangent  to  the  extrados-  at  the  l-eij  ;  each  arch 
of  the  same  thickness  throughout ;  r=:t]ie  lialf-pau  ; 
d—the  thichiess  at  the  hey ;  F'=thrust  of  elliptical 
arches  I  F^^thrust  of  segmental  arches. 


"r 

Rise  =.  i  tlie  sjmu. 

Rise  =  i  the  span. 

i 

1     d 

Thrnst  in 

Thrust  in 

d       1   Thrust  in 

Thrust  in 

\=K'. 

,4=.-. 

the  elliptic- 

the segmen- 

!•' 

1  +  —=    the  ellii>tic- 

the  sesinen- 

F' 

i 

al  arch  =  F' 
=  r^xC. 

tal  arch  =/' 

F 

^       al  arch  =  /'' 

till    i\Tv\\  =  F 

=r-^.C. 

f' 

1 

i  60 

1.02300 

0.09874 

0.10747 

0.920  1 

1.02§ 

0.09O75 

0.09840 

0.920 

50 

1.02760 

0.10628 

0.11334 

0.940 

1.0320 

0.09703 

0.10322 

0.940 

I  40     1.03448 

0.11687 

0.12169 

0.960 

1.04O0 

0.10600 

0.11021 

0.960 

,   30  1  1.04600 

0.133n8 

0.13479 

0.990 

1.05i 

0.11977 

0.12101 

0.990 

25     1.05520 

0.14469 

0.14450 

1.000 

1.0640 

0.13005 

0.12911 

1.010 

1   20     1 .06900 

0.16054 

0.15798 

1.020 

I.O80O 

0.14359 

0.14040 

1.020 

15   '  1.09000 

0.18355 

0.17610 

1.040 

1.103 

0.16349 

0.15692 

1.040 

10 

1.13800 

0.22374 

0.20880 

1.070 

1.16O0 

0.19564 

0. 18300 

1.070 

8 

1 . 2000 

0.21684 

0.19737 

1.100 

Rise  =  J  the  span. 

1 

Rise  =  i  the  span. 

60 

1.03077 

0.08238  ;  0.08478 

0.970' 

1.0320 

0.07943 

0.07905 

1^00 

50     1.03690 

0. 087 27 

0.08837 

0.990 

1.0384 

0.08386 

0.08250 

1.016 

40     1.04615 

0.09448 

0.09373 

1.010: 

1.0480 

0.09047 

0.08764 

1.030  1 

30     1.06150 

0.10527 

0.10222 

1.030 

1.0640 

0.10011 

0.O958O 

1.045 

25     1.07380 

0.113H6 

0.10869 

1.043 

1.0768 

0.10747 

0.10174 

1.056 

20  !  1.09230 

0. 12453 

0.11719 

1 .  O60 

1.0960 

0.11758 

0.10988 

1.070  , 

1   15   1  1.12300 

0.14065 

0.12999 

1.08<»' 

\  1.1280 

0.13225 

0.12194  • 

1.085 

10  1 1.18460 

0.16564 

0.14965 

1.110 

i 1.1920 

0.15474 

0. 14008 

1.100  1 

8   '  1.23077 

0.17977 

0.15966 

1 . 1 30 

! 1 . 2400 

0.16680 

0.14906 

1.120 

6  I  1.30770 

0.19848 

0.16872 

1.170 

1 

1 . 3200 

0.18082 

0.15690 

1.150 

102.  In  the  preceding  tiiLle  we  have  supposed  every 
arch  to  liave  a  constant  thickness  throughout. 

Ill  practice,  all  light  arches  should  increase  in  thickness 
from  the  key  to  the  springing  line,  as  already  exjJained. 
Such  increase,  a])]»lied  to  the  arches  of  the  preceding  tahle 
■will  diiiiiiii>h  tlie  thni-t^  of  elliptical  and  scgincutal  arches, 
without  sensibly  changing  their  relative  magnitudes. 


ELLIPTICAL    ARCHES.— SURCIIARG  EI)    HORIZONTALLY.       305 


EFFECT   OF   SURCHARGE    UPON    THE    ROTATION    THRUST. 

Fiijiire  22. 

103.  Suppose  a  surcliarge  of  tlie  density  of  tlie  arch 
and  of  the  uniform  vertical  depth  t:=ad.      On  the  anxil- 

iary  circular  arch  lay  off  a'  d'=~rXf.     Let  ns  take  into 

account  in  reference  to  any  joints  m  n^  m  n\  only  that  part 
of  the  surcharge  which  overlies  the  segments  upon  the 
right  of  the  vertical,  m  m'.  The  effect  of  the  surchai'ge 
will  be  precisely  the  same  in  the  two  arches.  But  the 
addition  to  the  thrust  caused  by  this  surcharge  in  the  cir- 
cular arch  will  be  A=-jtrj^- 7)  of  which  the  max- 

/     it— cos.'y 

imum  value  is  * 

The  following  table  gives  the  values  of  the  factor  (A^— 


■\/IP—l)  for  values  of  Pranging  from  1  to  1.40. 


306 


THEORY     OV    THE    ARCH. 


104.  Tahle  of  additions  to  the  rotation  thrust  caused  hy  a 
surcharge  of  constant  vertical  depth;  d=the  thickness 
of  the  elliptical  arch  at  the  ley  ;  r=tlie  ludf-span ;  f= 
the  rise ;  t=the  depth  of  the  surcharge  on  the  elliptical 
arch:  A— the  addition  to  the  thrust  of  the  •elliptical 
arch;  C—the  decimal  in  any  column ;  K—the  ratio 
of  the  two  radii  of  the  auxiliary  semicircular  arch. 

Wt-  hare  K=l^j ;  A^r'^XyXC. 

The  results  of  this  table  are  always  slightly  in  excess. 


1     t  1' 

A-r"-xKc.\ 

A=r^xlxC. 

JrrrSx-xa 

1     ''    " 

/    -I,   .  rf     K\ 

/        \ 

f 

/ 

f      r  \ 

=  A'. 

! 

K. 

K. 

=  K. 

c. 

a 

C. 

'•         I 

1.00 

1.00000  '   1.11 

0.62823 

1.22 

0.52114 '11.33 

0.45313 

1.01 

0.80823 

1.12 

0.61562  i 

1.23 

0.51383 

1.34 

0.44804 

1.02 

0.81900 i 

1.13 

0.60379 

1.24 

0.50679 

1.35 

0.44308 

1.03 

0.78322  ! 

1.14 

0.59264 

1.25 

0.50000 

1.36 

0.43826 

1.04 

0.75434  1 

1.15 

0.58211 

1.26 

0.49345 

1.37 

0.43357 

l.Oo 

0.72984 j 

1.16 

0.57212  : 

1.27 

0.48712',  1.38 

0.42900 

l.OC 

0.70843 

1.17 

0.50263 

1  1.28 

0.48100'   1.39 

0.42455 

1.07 

0.08934 

1.18 

0.55358; 

j  1.29 

0.47508     1.40 

0.42020 

'  1.08 

0.G7208 

1.19 

0.54494.1 

1.30 

0.46934: 

1  1.09 

0.65630 

1.20 

0.53668 

1.31 

0.46378  1 

li.io 

0.64174 

1.21 

0.52875 

1.32 

0.45837 

lOo.  Example.  The  Waterloo  Bridge  :' span  =  120  feet 
=27*;  rise=32  feet=/;  tliickne^s  at  the  key=4.r)0  teet=i(/. 
These  dimensions  give,  the  number  in  the  first  coluinu  of 


!!/• 


table  G,  7r  =  ^^=26|;  /=/'X0.53i 

Tliru^t,  3(1   coluiiiii  table   G,  for /r  =  26, 

/=rX0.50, i^=r^X  0.12767 

Ditto  for  /r'=28,  /=7'X0.50,     .         .       i' =  7-^X0.12347 


ELLIPTICAL     ARCHES    SURCHARGED     IIORIZOKTALLY.  307 

Subtracting  one  third   of  tlie  difference 

from  the  former,  we  liave,  for  7r'=:26|, 

/=rX.50, J^=r-X0.12G27 

In  like  manner  we  find  for  /r'  =  2G|-,/=: 

^r=;'X0.55,      ....         i^izi/-^x0.120GS 

Adding  one  third  of  the  difference  to  tlie 
latter  we  have  the  required  thrust,  cor- 
responding to  /r  =  26|,/=rX0.53L,   i^=y-x 0.12254 

Suppose  a  surcharge  4  feet  deep  through- 
out, we  find,  art.  104,  opposite  A"=l-f- 
/'Z  4  'lO  / 

J  ^'^  / 

X^X  0.59264,       ....         =/-'x  0.07408 
o2 


Total  thrust,  i^r=/-2x  0.19662 

=  707.83  ;  that  is,  the  thrust  upon  one  foot  in  width  of  the 
bridge  at  the  key,  is  equivalent  to  the  weight  of  a  column, 
of  the  material  of  the  bridge,  one  foot  square  and  707.83 
feet  high.  Dividing  this  l)y  the  thickness  of  the  arch,  4^ 
feet,  we  have,  as  the  mean  pressure  upon  each  square  foot 
of  surface  at  the  key,  157.30.  This  mean  pressure,  accord- 
inir  to  the  best  authorities,  should  never  exceed  one  twen- 
tietli  of  the  ultimate  strength  of  the  material. 

Example  II.     ?'=10' ;  /=6f  feet ;  ^=1' ;  giving  7^'= 

20;/=|r; 

Thrust,  6th  column  table  G,  opposite  K' 

=  20, i^=/-X0.12453 

Three-center  arch  of  the  same  rise,  span, 

and  thickness,  M.  Audoy,  .  .  F=  r  X  0. 1 3089 
Difference,  about  5  per  cent,  of  the  thrust,      =/•-  X  0.00636 


308  THEORY     OF    THE    ARCH 


SLIDING    TIIUUST    OF    ELLIPTICAL    AllCHES   SURCHAEGED 
HORIZONTALLY. 

lOG.  Tliis  thrust  will  always  he  le^^s  than  the  rotation 
thrust,  unless  the  arch  have  an  enormous  thickness  at  the 
key. 

For  reasons  substantially  the  same  with  those  given  in 
art.  94,  we  are  justified  in  offering  the  following  rule  as 
perfectly  safe,  but  liable  to  give  a  thrust  somewhat  too 
large. 

Rule.  Find  the  rotation  thrust  as  above  explained. 
Should  this  thrust  be  less  than  the  sliding  thrust  in  table 

D,  opposite  K=l-\--,  adopt  the  latter  as  the  true  thrust. 

ft/  ' 


THICKNESS    OF    PIER. 

107.  The  formulae  of  art.  64  are  api)licaTjle  to  all  cases. 
Api)lied  to  elliptical  arches  surcharged  horizc^ntally,  Ave 
have 

i/-2^-/yx0.4520G5;  ^  ^ 

ill  all  cases  ?=the  elevation  of  the  extrados  at  the  crown 
above  the  base  of  the  pier. 

Example.  Waterloo  Bridge,  dimensions  and  load  given 
in  art.  105.  Deptli  of  the  pier  below  the  springing  line  = 
19'.50;  A  =  GO';  ^=32'+4'.50  +  4'=40'.50;  n  =  QOX 
40.50  -  GO  X  32  X  0.7854  =  922.03  ;    m  =  K^O)' X  40.50- 

2 

(60yx32x0.452065=20822.     Z=56';  |=15.367;   j,= 
23G.15;  ^=347.03;  ^-''=GG0.G4;    F,  art.  105,  =707.83. 


ELLIPTICAL    ARCHES    SUUCIIAROED     IIOIMZONTALLY.         309 

These  values  siihstitiited  in  (31),  art.  64,  give,  for  strict 
equilibrium,  5  =  1,   .....         e=14'.02 
(31),  art.  64,  give,  for  c^=:2,      ....     6^=31'.37 
(31i),     "  "       x=ie,         .         .         .         t^=22'.43 

(31|),     "  "       a?=:|6^    ....     e=z2(j'A^ 

The  thickness  of  the  existing  jitiers  is,  at  the  bottom 
30',  at  the  S2:)ringing  line  20',  and  the  piers  extend  al)Ove 
and  below  the  bridge  about  one-fourth  the  width  of  the 
bridge,  eacli  way.  Every  pier,  therefore,  of  this  cele- 
brated bridge,  is  an  al)utinent  pier,  with  very  nearly  the 
excess  of  stability  prescribed  by  the  French  engineers. 
Comparing  the  moment  of  the  thrust  with  the  sum  of  the 
moments  of  all  the  elements  of  resistance,  we  find  5,  the 
coefficient  of  stability,  to  be  very  nearly  1.79. 


I 


310 


THEORY     OF    THE     ARCH. 


Circular  Arches  of  180°,  with  Paiiallkl  ExrR.\DOS. 

(A)   Talle  gicing  the  anfjle  of  rupture^  the  thru-^f,  cuhI  the 
Ii)iiit  thicl'ne-96-  of]}fer-9. 


!'  V, 

Value 



1 

Ratio  V2.  C.  of  the  limit  thick- , 

'     Value 

Ratio 

of  the 

Ratio  C,  of  the  thrust  to   the' 

less  of  Pier  to  the  radius  <f  I  he 

;    ..fthe 

of  the 

An^rle      s 

quari'  of  the  rad 

US  r.  of  the  in- 

ntrados. 

rati" 

diameter 
to  the 

of 
Rupture. 

Lrados. 

K=^ 

C'o-cfflcicnt  of  1 

thickness. 

Strict           i 

Rotation. 

Rotalion.        | 

Sliding. 

Equilibrium.    | 

lability,  1. 90 

2.732 

1.154 

(j°  0(J' 

O.OI  111(10 

0.98923 

2.70 

1.176 

13  42 

0  (102 11 

0.96262 

!     2.05 

1.212 

22  00 

0.0(1319 

0. 02168 

1     2.60 

1.25t) 

27   30 

0.00808 

0.88151 

[     2 .  50 

1.333 

35  62 

0.02283 

0.8O346 

1     2.40 

1.428 

42  06 

0.04 109 

0.72847 

!     2.30 

1.538 

46  47 

0.O68S5 

0.65654 

i     2 .  20 

1.666 

51   04 

0.08(548  • 

0.58767 

2.10 

1.810 

54  27 

0.10926 

0.52186 

2.00 

2 .  000 

57    17 

0.13(il7 

0.45912 

0.9582 

1.3223          1 

1.90 

2.282 

59  37 

0.14813 

0.39943 

0.8938 

1.2320          i 

1.80 

2.500 

61   24 

0.1t>373 

0.34281 

0.8280 

1.1414         ji 

'     1.70 

2.857 

62  53 

0.17180 

0.28924 

0  7606 

1.0484 

1.60 

3.333 

63  49 

0.17517 

0.23874 

0.6910 

0.9525         1 

1.59 

3.389 

63  52 

0.17533 

0 . 23386 

0.6839 

0 . 9427          , 

i 

1.58 

3.448 

63  55 

0.17535 

0.229-11 

0.6768 

0 . 9329        !  1 

! 

1.57 

3.508 

63  58 

0.17524 

0.22434 

0.6698 

0.9233        ;l 

1.56 

3.571 

64  01 

0.17499 

0.21940 

0.6624 

0.9131         1 

1 

1.55 

3.636 

64  03 

0.17478 

0.21464 

0.6552 

0.9031 

1 

1.54 

3.703 

64  05 

0.17445 

0.20991 

0.6479 

0.8931          1 

'     1.53 

3.773 

64  07 

0.17397 

0.20521 

0.64(J6 

0.8831          1 

1     1 .  52 

3.846 

64  08 

0.17352 

0.200.04 

0.6333 

0.8730 

'     1.61 

3.920 

64  08 

0.17310 

0.19.590 

0.62.^9 

0.8628         1 

1.50 

4.000 

64  09 

0.17254 

0.19130 

0.6185 

0 . 8527         ;  j 

1.49 

4.0-1 

64  08 

0.17180 

0.1 867 3 

0.6111 

0.8424        ' 

j 

1.48 

4.166 

64  08 

0.17095 

0.18218 

0.6(136 

0 . 8320        1 , 

1.47 

4.255 

64  07- 

0.170(18 

0.17766 

0.5961 

0.8216 

1.46 

4.347 

64  06 

0.16915 

0.17318 

0 . 5885 

0.8112 

1.45 
1.44 
1.43 

4.444 
4.545 
4.651 

6t  05 
64  03 
64  00 

0.16798 
0.16683 

0.1*6872 

0.5809 

0.8007 

0.16430 
0.15991 

0.6776 
'        0  5756 

0.7962 
0.7934 

0.16568 

1.42 

4.761 

63  66 

0.16448 

0.155.1 5 

0.5735 

0.7906 

1.41 

4.878 

63  62 

0.16317 

o.l.il22 

0.5713 

0.7874 

1 

1.40 

5.000 

63  48 

0.16167 

0.14691 

0.5686 

0.7838        11 

1.39 

6.128 

63  43 

0.16014 

0.14264 

0.5659 

0.7801        j' 

1.38 

6.263 

63  38 

0.15846 

0.13841 

0.5629 

0.7760 

1.87 

6.406 

63  32 

0.15672 

0.13420 

0.6598 

0.7717 

! 

TABLES. 


Circular  Arches  of  180",  with  Parallel  Extrados. 

(A)   Table  giving  tlie  angle  of  rvpture,  tlie  thrust,  and  the 
limit  tldclaiess  of  piers. 


Viiluo 

Eatio  \^i.  a  of  the  limit  thick- 

Value 

E:vtio 

of  the 

Eatio,  C.  of  the  thrust  to  the 

ness  of  pier  to  the  radius  of  the  1 

of  the 

of  the 

angle 

square  of  the  radius,  ?',  of  the  in-^ 

intrados. 

ratio 
r 

diameter 

to  tbe 
thickness. 

of 
rupture. 

trades. 

Strict 

Co-effioieiit  of 

' 

\ 

Kotation. 

Rotation.       ] 

Slidinsc 

Equilibrium. 

stability  1.90. 

1.36 

5.565 

63°  26' 

0.15482 

0.13002 

0 . 5564 

0.7670        1 

1.35 

5.714 

63   19 

0.15287 

0.12587 

0.5529 

0.7622 

1.84 

5 .  882 

63   10 

0.15096 

0.12176 

0.5495 

0.7574 

1.33 

6 .  060 

63  00 

0.14896 

0.11767 

0.5458 

0.7524 

1.32 

6.264 

62  50 

0.14678 

0.11362 

0.5418 

0.7468 

1.31 

6.451 

62  33 

0. 14510 

0.10959 

0.5387 

0.7425 

1.30 

6.666 

62  14 

0.14330 

0.10559 

0.5353 

0.7379 

1.29 

6.896 

62  09 

0.14013 

0.10163 

0.5294 

0.7297 

1.28 

7.142 

62  03 

0.13691 

0.09770 

0.5233 

0.7213 

1.27 

7.4(i7 

61  47 

0.18430 

0.09379 

0.5183 

0.7144 

1.26 

7 .  692 

61  30 

0.13157 

0.08992 

0.5130 

0.7071 

1 .  25 

8.000 

61   15 

0.12847 

0.08608 

0.5069 

0.6987 

1.24 

8.333 

61  01 

0.12516 

0.08-i27 

0.5O()3 

0.6896        i 

1.23 

8.695 

60  40 

0.122(11 

9.07849 

0.4940 

0 . 6809        ! 

1.22 

9.090 

60  19 

0.11887 

0.07474 

0.4876 

0.6721 

1.21 

9.523 

60  00 

0.11516 

0.07102 

0.4799 

0.6615 

1 .  20 

10.000 

59  41 

0.11140 

0.06733 

0.4720 

0.6504 

1.19 

10.526 

59  10 

0.10791 

0.06368 

0.4646 

0 . 6404 

1.18 

11.111 

58  40 

0.10417 

0.06005 

0.4564 

0 . 6292 

1.17 

11.764 

58  09 

0.10021 

0.05646 

0.4472 

0.6171 

1.16 

12.500 

57  40 

0.09593 

0.05289 

0.4380 

0 . 6038 

1.15 

13.338 

57  01 

0.09176 

0.04935 

0.4284 

0.5905 

l.U 

14.285 

56  23 

0.08729 

0.04585 

0.4178 

0.5759 

1.13 

15.384 

55  45 

0.08254 

0.04237 

0.4063 

0.5601        1 

1.12 

16.666 

54  48 

0.07789 

0.03984 

0.3947 

0.5144        j 

l.U 

18.181 

54  10 

0.07273 

0.03552 

0.3814 

0.5259        1 

1 .  10 

20.000 

53  15 

0.06754 

0.03213 

0.3675 

0.5066 

1.09 

22.222 

52  14 

0.06177 

0.02879 

1.08 

25.000 

51  07 

0.05649 

0.02546 

1.07 

28.571 

49  48 

0.05065 

0.02217 

1.06 

33.333 

48   18 

0.04455 

0.01891 

1 

1.05 

40.000 

46  32 

0.03813 

0.01568 

1 

1.04 

50.000 

44  04 

0.03139 

0.01249 

1 

1.03 

66.666 

41   04 

0.02459 

0.00932 

i 

1.02 

100.000 

38   12 

0.01691 

0.00618 

1.01 

200.000 

32  36 

0.00889 

0.00308 

1.00 

Infini. 

0  00 

0.00000 

0.00000 

ClECULAE    AkCIIES    OF    ISO' ExTRADOS    AXD    IXTRADOS 

Parallel. 
(B)  TaUe  of  Thichness  of  Piers. 


Value 
of  thr 
ratio 

r 


2.00 

1.90 

1.80 

1.70 

1.60 

1.59 

1.58 

1.57 

1.5G 

1 .55 

1.54 

1 .5<j 

1.52 

1.51 

1.50 

1.49 

1.48 

1.47 

1.4ij 

1.45 


Katioof  i     Kat'o  -  of  the  thickness  of  piers  to  the  radius  of  the  intrados, 

the       I  *■  ,. 

diameter  in  function  of  the  ratio  -.  of  this  radius,  to  the  height  of  the 

to  the    !  '« 

thickness  piers.  case  ok  steict  equilibrium. 


2.000 

0    <)')<) 


3.389 
3.448 


3.773 
3.846 


44 

4:i 

1 

42 

1   ^ 

.41 

.40 

.39 

i 

.38 

—2.3562-+  t  (5.5517- +  1.7907 -  +  0.9182) 

—2.0449^  +  1  (4.2021  75  +  1.3240^+  0.79SS) 

0.500  i  —1.7593  T-  -I-  1  (3.0951  v.  +  0.9368*-  +  0.685C) 

2.857   '  —1.4844  ^  +  tf2.2tt34  I7.  +  0.6933^  +  0.5785) 
I  h  It'  II 

3.333     —1.2252  r  +  i  (1.5012  7^  +  0.3775 'i-  +  0.4775) 
//  ^  Ir  h 

r  r''  r 

—1.2001  r  -  t  .1.4404-,  +  0.36667-  +  0.4677) 
/(  /i  li 

— 1.17.52  r  4-  s  (1.3812'-,+  0.3361^  +  0.4580) 
3.508     —1.1513^+  \  (1.3255^,  +  0.3151  ^  +  0.4187)  j; 
3.571   i— 1.1261  r-i-  »  (1.2677  ,-3+  0.2966  7-  +  0.4388)  ;| 

3.636     — l.lnlo^^  »  (1.2133^,  +  0.2783^-  + 0.4293)  Ij 

I  ,'  r-"  r  II 

3.703  I  —1.07727-  +  1  (1.1605  7-2+  0.2603  T-  +  0.4198) 

\  li  h  ft 

—1.0631  5-+  V  (1.1091  ^.,+  0.2428^  +  0.4104) 

— 1.0292 '--t-  \  (1.0592^^-,  +  0.2224^  +  0.4011) 

3.920  1—1.0073^+  »  (1.0146 ',i+  0.2056'-+  0.3918) 
h  Ir  li  •    =^ 

4.000     —0.9817  7-+  1  (0.9638^+  0.1937  y^-  +  0.3826) 

4.081   1—0.9583^  +  t  (0.9184 '^  +  0.1684  ^  +  0.3735)  |l 

r"  r 

4.166     —0.9349^  +  t  (0.8741  ^5  +  0.1659^+  0.3644; 

4.255  ;  -0.9125-  +  i  (n.8328  7-„+  0.1482^+  0.3553) 

4.347  1—0.8887^+  1  (0.7899^^  +  0.1362^  +  0.3464) 

4.444  1— f>.8659^+  V  (0.7498  7i+  0.1232^  +  0.3374) 
i  h  li  li 

I  J-  ?•'  »• 

4.545  1—0.8432-  +  »  (0.7110  7,+  0.1181  ,-  +  0.3337) 
li  h  It 

—0.8206^  +  t  (i).6735Va  +  0.1163 '^-  +  0.3314) 

—0.7983^+  ,  (0.6372'    +  0.1143  7-  +  0.3290) 
h  li  11 

4.878  !— 0.7760  1  +  ,  (0.6023'- +  0.1102*-  +  0.3263) 
1  /(  Ir  li 

5.000  '  —0.7540  ^  +  t  (0.5685  7-,  +  0.1074  7    _(-  0.3233) 
//  Ir  It 

5.128  i— 0.7321  7-+  »'(0.5359  7,  +  0.1048  7- +  0.3203) 
h  Ir  H 

5.263     —0.7103^+  i  (0.5045^. +  0.1021  ^  +  0.3169) 
I  li  Ir  n  


4.65V 
4.761 


Circular  Arches  of  180°— Extrados  and  Intrados 

Parallel. 
Continuation  of  Tahle  (B).     Thickness  of  Piers. 


Value 
of  the 
ratio 


1.37 

1.36 

1.35 

1.34 

1.33 

1.32 

1.31 

1.30 

1.29 

1.28 

1.27 

1.26 

1.25 

1.24 

1.23 

1 .22 

1.21 

1.20 

1.19 

1.18 

1.17 

1.16 

1.15 

1.14 

1.13 

1.12 

1.11 

1.10 


Ratio  of 

tho 
clianietcr 

to  the 
thickness 


Eatio  -  of  tho  thickness  of  piers  to  tho  radlusof  tho  intrados, 

in  function  of  tlio  ratio  i',  of  tills  radius,  to  the  hei^'ht  of  the 
A 


5 .  406 
5.555 
5.714 
5.882 

6 .  060 
6.264 
6.451 
6.666 
6.896 
7.142 
7.407 
7.692 
8.000 
8.333 
8.695 
9.090 
9.523 

10.000 
10.526 
11.111 
11.764 
12.500 
13.333 
14.285 
15.384 
16.666 
18.181 
20.000 


CASK   OF  STRICT   E<Jt,'ILIHRIU.Nf. 


—0.6887  "■-  +  t/ (0.4743  \.^  +  0.0995^  +  0.3134) 

—0.6673^  +  t/(0.4452  J^-^  +  0.0969^  -f  0.3096) 

—0.6460  ~  +  V (0.4173  ^  +  0.0944^  +  0.3057) 

—0.6249^+  ^'(0.3904  7,^  +  0.0926  y-  +  0.3019) 
It  h'  h 

—0.6050-  +  i'(0.3660',^  +  0.0903^  +  0.2979) 

—0.5831  ^  + v(0.3400^,  +  0.0880  ~  +  0.2936) 

— 0.5G24-  -I-  v'(0.3163^,  +  0.0875*^-+-  0.2902) 

—0.5419  ~  +   v'(0.2937  ''-^  +  0.0867  ^  +  0.2866) 

—0.5216  7-  +  V(0.2720^,  +  0.0828^  +  0.2803) 
/'  It-  I  I 

—0.5014  ^  +  V (0.2520^^  +  0.0801  j  +  0.2738) 

—0.4926  ^  +  V (0.2426  '-'  +  0.0778  r  +  0.2686) 

—0.4616  ^  +  V(0.2130  ''-  +  0.0755  ^  +  0.2631) 

—0.4418  ~  +  1(0.1952  ~  +  o.0730^  +  0.2569) 

—0.4222  ~  +  V(0.1783  -^  +  0.0713  j  +  0.2503) 

— 0 .  4028  "^  +  V  (0.1623^  +  0. 0684  ^  +  0 .  2440) 

—0.3836'-+  1(0.1471^+  0.0674^  +  0.2377) 

— 0 .  3645  ^  +  V  (0 .  1329  ^  +  0 .  0641  ^  +  0 .  2303) 

—0.3456^  +  4'(0.1194^  +  0.0614^^  +  0.2228) 

—0.3268^  +  v'(0.1068^+  0.0600^  +  0.2158) 
h  It  h 

—0.3082  ~  +  V (0.0950  ^^  +  0.0581  ~  +  0.2083) 

—0.2897 '7-  +  l'(0.0840^  +  0.0561  7-  +  0.2004) 
h  h'  I  I 

—0.2714^+  v'(0.0734^j  +  0.0559^  +  0.1919) 

—0.2533^-+  1' (0.0642^,  +  0.0536^  +  0.1835) 

—0.2353^  +  V(0.0554^+  0.0513^-  +  0.1745) 

—0.2175^+  ^'(0.0473*-  +  0.0490^  +  0.1651) 
/(  //■'         h 

r     •        r*         r 
—0.1998^+  t'(0.0399- +  0.0467  t- +  0. 1557) 

—0.1823^  +  V(0.0332^-i-  0.0426^  +  0.1455) 

— 0.1649  r+  1'(0.0272^„-i-  0.0394^  +  0.1351) 
n  Ir  h  ' 


3U 


THEORY    OF    THE    ARCH. 


Circular  Arches  of  180^  avith  a  Surcharge  in 
Masonry  inclined  45°  on  each  side  of  the  Cen- 
tral Ridge. 

(C)  Table  giving  the  angle  of  rupture,  the  thrust,  and  the 
limit  thickness  of  piers. 


\   Value 
1    of  the 

Eatio 
of  the 

Value 
of  the 

EaHo,  C,  of  th 
square  of  the  rad 

e  thrust  to  the 
us,  r,  of  the  in- 

Ratio  y/'l.  C.  of  the  limit  thick- 
ness of  pier  to  the  radius  of  the 
intrados. 

ratio 

1             ! 

diameter 

to  the 
thickness. 

angle  of 
rup  ure. 

Rotafn.i 

tradus. 

Strict 
Equilibrium. 

Coefficient  of 
gtability  2. 

Rotation. 

Sliding. 

i  2.(10 

2.000 

60° 

0.26424 

0.74361 

1.2212 

1.7246 

'   1.90 

2.222 

60 

0.28416 

0 . 65648 

l.i458 

1.6204 

I  1.80 

2.500 

60 

0.29907 

0.57383 

1.0759 

1.5147 

:  1.70 

2.857 

60 

0.30867 

0.49564 

0.9956 

1.4081 

i  1.60 

3.333 

60 

0.31245 

0.42191 

0.9186 

1.2990 

1.59 

3.389 

60 

0.31249 

0.41478 

0.9108 

1.2880 

'  1.58 

3.448 

60 

0.31257 

0.4(^841 

0.9038 

1.2781 

!  1.57 

3.508 

61 

0.31264 

0.40067 

0.8952 

1.2660 

1.56 

3.571 

61 

0.31246 

0.39367 

0.8864 

1 . 2548 

1.55 

3.636 

61 

0.31222 

0.38673 

0.8795 

1 . 2437 

1.54 

3.703 

61 

0.31191 

0.37983 

0.8716 

1.2318 

1.53 

3.773 

61 

0.31153 

0.37297 

0.8637 

1.2214 

1.52 

3.846 

61 

0.31108 

0.30615 

0.8557 

1.2102 

1.51 

3.920 

61 

0.31056 

0.35938 

0.8478 

.1.1989 

1.50 

4.000 

61 

0.30996 

0 . 35266 

0.8398 

1.1877 

1.49 

4.081 

61 

0.30928 

0.34.598 

0.8318 

1.1764 

;    1.48 

4.166 

61 

0 . 30855 

0.33934 

0.8238 

1.1650 

1.47 

4.255 

61 

0.30772 

0.33275 

0.8158 

1.1537 

'    1.46 

4.347 

60 

0.30685 

0.32621 

0.8077 

1.1422 

1.45 

4.444 

60 

0.30587 

0.31971 

0.7996 

1.1308 

1.44 

4.545 

60 

0.30485 

0.31325 

0.7915 

1.1193 

1.43 

4.651 
4.761 
4.878 

60 
60 
60 

0.30408 
0.30296 

0.30684 

0.7834 

1.1078 

1.42 
1.41 

0.30047 

0.7784 
0.7768 

1.1008 
1.0986 

0.30173 

1.40 

6.000 

59 

0.30001 

0.28787 

0.7746 

1.0954 

1 .  39 

5.128 

59 

0.29712 

0.7709 

1.0914 

.    1.38 

5.263 

59 

0.29706 

0.7690 

1.0914 

1    1.37 

5.406 

59 

0.29550 

0.7688 

1.0872 

TABLES. 


315 


Circular  Arches  of  180°,  with  a  Surcharge  in 
Masonry  inclined  45°  on  each  side  of  the  Cen- 
tral KiDGE. 

(C)   TaUe  giving  the  angle  of  rtiptiire^  the  thrust^  and  the 
limit  thickness  of  piers. 


i 

^     .   ^     .V      X  ^    ^T-   1     Ratio  V'2.  C.  of  the  limit  thick-' 

Value 

Ratio 

Value 

Ratio    C  of  the  thrust  to  the  n^g,  „f    jg,  ^o  t 
square  of  the  radius,  r,  of  the  in-  intrados. 

le  radius  of  the 

of  the 

of  the 

of  the 

1 

ratio 
r. 

diameter 

to  the 
thiclcness. 

an^le  of 
rupture. 

trados. 

11 

Strict 

Coefficient  of 

1 

Eotafn. 

Rotation. 

Sliding.             Equilibrium. 

stability  2. 

1.36 

5.555 

59° 

0.29386 

0.7665 

1.0841  • 

1.35 

5.714 

58 

0.29285 

0.7653 

1.0823 

1.34 

5.882 

58 

0.29037 

0.7621 

1.0777 

1.33 

6.060 

58 

0.28850 

0.7596 

1.0742 

1.32 

6 .  264 

58 

0.28654 

0.7570 

1.0705 

1.31 

6.451 

67 

0.28456 

0.7544 

1.0668 

1.30 

6.666 

57 

0.28231 

0.22756 

0.7514 

1.0626 

1.29 

6.896 

57 

0.28027 

0.7487 

1.0588 

1 .  28 

7.142 

56 

0.27810 

0.7458 

1.0547 

1.27 

7.407 

56 

0.27578 

0.7427 

1.0503 

1.26 

7.692 

55 

0.27343 

0.7395 

1.0458 

1.25 

8.000 

54 

0.27102 

0.7362 

1.0412 

1.2-1 

8 .  333 

53 

0.26850 

0.7328 

1.0363 

1.23 

8.6S5 

53 

0.26608 

0.7274 

1.0316 

1.22 

9.090 

52 

0.26377 

0.7263 

1.0272 

1.21 

9.523 

51 

0.26074 

0.7221 

1.0217 

1.20 

10.000 

50 

0.25806 

0.17172 

0.7184 

1.0160 

1.19 

10.526 

50 

0.25546 

0.7148 

1.0109 

1.18 

11.111 

49 

0.25277 

0.7111 

1.0045 

1.17 

11.764 

49 

0.25010 

0.7072 

1.0002 

1.16 

12.500 

48 

0 . 24742 

0.7034 

0.9948 

1.15 

13.333 

47 

0.24477 

0.6997 

0.9894 

1.14 

14.285 

46 

0.24218 

0.6960 

0 . 9842 

1.13 

15.384 

44 

0.23967 

0.6923 

0.9791 

1.12 

16.666 

43 

0.23732 

0.6889 

0.9743 

1.11 

18.181 

43 

0 . 23502 

0.6856 

0.9695 

1.10 

20.000 

42 

0.23292 

0.12032 

0.6825 

0.9652 

1.05 

40.000 

36 

0 . 22902 

0.6768 

0.9571 

316 


THEORY     OF    THE     ARCH. 


Circular  Arches  of  180°,  Loaded  tp  to  the  Level 
OF  THE  Top  of  the  Key. 

(D).  Tahle  giving  the  angle  of  ritptvre^  the  thrust,  and  the 
limit  thicl'nes-9  of  piers. 


Value 

Ratio 

Value 

Ratio,  C,  of  the  thrust  to  the 

Ratio  of  the  limit  thickness  of 

of  the 

of  the 

of  the 

square  of  the  radius,  r,  of  the  in- 

pier  to  the  radius 

of  the  intrados. 

Katio. 

diameter 
to  the 

angle  of 
rupture. 

trados. 

1 

^ 

Strict 

Coefficient  of 

K=— 

thickness. 

1 

r 

Rotat'n.  1 

Rotation.       { 

Slidiug. 

Equilibrium. 

stal.ility  1.00. 

2.00 

2.000 

36° 

0.05486 

0.5(1358 

1.0036 

1.3834        1 

1.90 

2 .  222 

39 

0.071U1 

0.43966 

0.9377 

1.2925 

1.80 

2.500 

44 

0.08850 

0.37901 

0.8706 

1.2(J0l 

1.70 

2.857 

48 

0.10631 

0.32164 

0 . 8020 

1.1055 

1.60 

3.333 

52 

0.12300 

0.26755 

0.7315 

1.0082 

1.59 

3.389 

52 

0.12453 

0.26232 

0.7243 

0.9984 

1.58 

3.448 

53 

0.12602 

0.25712 

0.7171 

0.9885 

1.57 

3.508 

53 

0.12747 

0.25196 

0.7099 

0.9784 

1.56 

3.571 

54 

0.12837 

0.24683 

0.7026 

0.9684 

1.55 

3.636 

54 

0.13027 

0.24173 

0.6953 

0.9584 

1.54 

3.703 

55 

0.  LSI  53 

0.23667 

0.6880 

0.9483 

1.53 

3.773 

55 

0.18289 

0.23163 

0 . 6806 

0 . 9381 

1        A   .  V  .# 

1.52 

3.846 

55 

0.18414 

0 . 22664 

0.6732 

0.9280 

1.51 

3.920 

55 

0.13531 

0.22167 

0.6658 

0.9177 

1.50 

4.000 

56 

0.13648 

0.21673 

0.6583 

0.9075 

1.49 

4.081 

56 

0.13756 

0.21183 

0.6509 

0.8972 

1.48 

4.166 

56 

0.13856 

0.20696 

0.6433 

0.8868 

1.47 

4.255 

57 

0.13952 

0.20213 

0.6358 

0.8764 

1.46 

4.347 

57 

0.14041 

0.19733 

0.6282 

0.8659 

1.45 

4.444 

57 

0.14122 

0.19256 

0.6206 

0.8554 

1.44 

4  545 

58 

0.141Vt5 

0.18782 

0.C129 

0.8448 

1.43 

4.651 

58 

0.14268 

0.18312 

0 . 6052 

0.8341 

1.42 

4.761 

58 

0.14311 

0.17845 

0.5974 

0.8234 

1.41 

4.878 

59 

0.14376 

0.17381 

0.5S06 

0.8126 

1.40 

5.000 

59 

0.14421 

0.16920 

0.5817 

0.8018 

:  1.39 

5.128 

59 

0.14456 

0.16463 

0.5738 

0.7909 

'1  1.38 

5.263 

59 

0.14481 

0.16009 

0.5658 

0.7799 

1.37 

5.406 

60 

0.14498 

0.15558 

0.5578 

0.7689 

1.36 

5.555 

60 

0.14506 

0.15111 

0.5497 

0.7577 

1.35 
1.34 
1.33 

5.714 
6.882 
6.060 

60 
60 
61 

0.14504 
0.14491 
0.14467 

0.14666 

0.5416 
0.5383 

0.7465 

0.14225 

0.7420 

0.5379 

0.7414 

1.32 

6.264 

61 

0.14460 

0.5377 

0.7412 

TABLES. 


317 


CiKCULAR  Arches  of  ISO,  Loaded  up  to  the  Level 
OF  the  Top  of  the  Key. 


(D)    Table  giving  the  angle  of  rupture^  the  thrust^  and  tlie 
limit  thichness  of  piers. 


Value 

Ratio 

Value 

Ratio,  (7,  of  the  thrust  to  the 

Ratio  of  the  limit  thickness  of 

of  tlie 

of  the 

of  the 

square  of  the  radius,  r,  of  the  in- 

)ier  to  the  radius 

of  the  intrados 

Ratio 
It 

diameter 
to  the 

angle  of 
nipture. 

trados. 

' 

A=— . 

thickness. 

^ 

Strict 

Coeffloient  of 

V 

Rotat'n. 

Rotation. 

Sliding. 

Equilibrium 

stability  1.90. 

1.31 

6.451 

61° 

0.143900 

0.5358 

0.7394 

1.30 

6.666 

61 

0.143320 

0.12495 

0.5354 

0.7379 

1 .  20 

6.896 

61 

0.142640 

0.5341 

0.7362 

1.28 

7 .  142 

62 

0.141860 

0.5326 

0.7342 

1.27 

7.407 

62 

0.141010 

0.5310 

0.7320 

1.26 

7.692 

62 

0.139880 

0.5289 

0.7290    ; 

1.25 

8.000 

62 

0.138720 

0.10405 

0.5267 

0.7260 

1.24 

8.333 

62 

0.137370 

0.5235 

0.7225        j 

1.23 

8.695 

63 

0.135930 

0.5214 

0.7187        1 

1.22 

9.090 

63 

0.134370 

0.5184 

0.7145 

1.21 

9.523 

63 

0.132630 

0.5150 

0.71)90 

1.20 

10.000 

63 

0.130730 

0.08397 

0.5113 

0.7048 

1.19 

10.526 

63 

0.128700 

0.5073 

0 . 6993 

1.18 

11.111 

63 

0.126500 

0 . 5030 

0.6933 

1.17 

11.764 

64 

0.124150 

0.4983 

0.6868 

l.Ki 

12.500 

64 

0.121820 

0.4936 

0.68113 

1.15 

13.333 

64 

0.118950 

0.06471 

0.4877 

0.6723 

1.14 

14.285 

64 

0.116080 

0.4818 

0.6641 

1.13 

15.384 

64 

0.113030 

0.4755 

0.6553 

1.12 

16.666 

64 

0.109790 

0.4686 

0.6459 

1.11 

18.181 

65 

0.106410 

0.4613 

0.6358 

1.10 

20.000 

65 

0.102790 

0.04627 

0.4535 

0.6249 

1.09 

22.222 

66 

0.098992 

0.4449 

0.6133 

1.08 

25.000 

66 

0.094967 

0.4358 

0 . 6007 

1.07 

28.571 

67 

0.091189 

(1.4270 

0.6886 

1.06 

33.333 

68 

0.086376 

0.4156 

0.5729 

1.05 

40.000 

69 

0.081755 

0.02865 

0.4044 

0.5573 

1.04 

50.000 

70 

0.0768.57 

1.03 

66.666 

71 

0.071853 

1.02 

100.000 

73 

0.066469 

1.01 

200.000 

74 

0.061324 

1.00 

75 

0.055472 

0.01185 

31S 


THEORY  OF  THE  ARCH. 


Segjiextal  Arches — Extrados  asd  Intrados  Parallel. 

(E)    Table  of  thrusts  ill  seven  s^jstems  ;  s=tJie  sjycoi  ;  f= 
the  rise ;  C=the  decimal  in  anycoliunn  ;  F—the  thrust 

^r'-a 


The  thrust  =  the  decimal  x  the  square  of  the  radius  of  the  intrados. 


Value 

I  of  the 

ratio 


«=5/. 
29, 


«=6/. 
r=5/. 


63. 


r  =  58°730".  r=:43''3610"  t)=36°52  10•■r=81'58■26■j«=2S'4•20• 


«=10/. 
r=13/. 

v=22°srw 


1.40     0.15445      0.14G91       0.14691 


35 

34 

33 

32 

,1 

31 

1 

30 

1 1 

.29 

28 

1  \ 

.27 

1 

.26 

1.25 

1 1.24 

1.23 

1.22 


!   1.21 

1 
1 .  20 

i.iy 

1.18 

,1.17 

'1.16 

1.15 

1.14 

;  1.13 

;  1.12 

,  1.11  1 

:,1.10 

;  i.ou 

1,1.08 

|1.07 

lll.Ofi 

1.05 
1.04 
1.03 
1  .02 
1.01 


0.14717 
0.14543 
0.143G4 
0.14173 
0.13976 

0.13764 
0.13543 
0.13311 
0.13068 
0.12815 

0.12547 
0.12270 
0.12(131 
0.11675 
0.11354 

0.11023 
0.10676 
0.10313 
0.09934 

0.09537 
0.09123 
0.08690 
0.08238 
0.07764 
0.07269 
O.OC737 
0.00211 

0.05636 
0.05062 
n  04431 
0.03776 
I  0.03096 
0.02378 
0.01(>25 
0.00834 


0.13U3O 
0.12987 
0.12781 
0.12634 
0.12486 

0.12331 
0.12164 
0.11988 
0.11803 
0.11609 

0.11402 
0.11251 
0.10958 
0.10725 
0.10460 

0.10196 
0.09915 
0.09617 
0.09303 

0.08975 
0.08634 
0.08257 
0.07869 
0.07459 
0.07042 
0.06563 
0.06077 

0.05652 
0.05U11 
0.04428 

o.o:!Soi 

0.03144 
0.(rJ437 
O.olGSl 
0.00871 


0.12587 
0.12171 
0.11767 
0.11362 
0.10959 


0.10682 
0.10563 
0.10437 
0.10304 
0.10160 

0.10009 
0.09850 
0.09679 
0.09499 
0.09305 

0.09102 
0.08885 
0.08653 
0.08408 

0.08144 
0.07866 
0.07568 
0.07251 
0.06911 
0.06548 
0.06158 
0.05739 

0.05288 
0.04804 
0.04280 
0.03709 
0.03095 
0.02424 
o.oicito 
0.00886 


0.14691 

0.12587 
0.12171 
0.11767 
0.11362 
0.10959 

0.10559 
0.10163 
0.09770 
0.09379 
0. 08992 


0.08668 
0.08549 
0.08423 
0.08291 
0.08148 

0.07999 
0.07834 
0.07651 
0.07468 

0.07204 
0.070.00 
0.06812 
0.00558 
0.06297 
0.06O26 
0.05666 
0.05345 

0.04934 
0.04426 
0.04058 
0.03550 
0.02992 
0.02309 
O. 01073 
o.o(»889 


0.14691 

0.12587 
0.12171 
0.11767 
0.11362 
0.10959 

0.10559 
0.10163 
0.09770 
0.09379 
0.08992 

0.08608 
0.08227 
0.07849 
0.07474 
0.07102 


0.06981 
0.06859 
0.00727 
0.06583 

0.06420 
0.06259 
0.O0O77 
0.05890 
0.05659 
0.05421 
0.05160 
0.04871 

0.04552 
0.04200 
0.03861 
0.03357 
0.02862 
0.02293 
0.01640 
0.00885 


0.14478 

0.12405 
0.11999 
0.11596 
0.11196 
0.10800 

0.10406 
0.10016 
0.09628 
U. 09244 
0.08862 

0.08483 
0.08108 
0.07735 
0.07366 
0.06999 

0.06636 
0.06275 
0.05918 
0.05212 


0.05004 
0.04904 
0.04803 
0.04671 
0.04451 
0.04384 
0.04214 
0.04023 

0.03806 
0.03560 
.0.03276 
0.02944 
0.02501 
0.02131 
0.01540 
0.00862 


«=16/. 
r=32.5/. 
v=U'  15'. 


0.07180 
0.06862 
0.06547 
0.06234 
0.05924 

0.05016 
0. 05311 
0 .  05008 
0.04709 

0.04411 
0.04116 
0.03824 
0.03534 
0.03217 
0.02902 
0.O2081 
0.02401 


0.02192 
0.02111 
0.02002 
0.01 8S2 
0.01720 
0.01524 
0.01199 
0.00747 


f 


TABLES. 


if^' 


^^ 


0  ty 


319 


k^ 


Table  E'. 

SEG:\rE]srTAL  Abcties  Loaded  up  to  the  Level  of  the 

Summit  of  the  Key. 

Tahle  of  tlirust-s  in  seven  systems. 


Multiply  the  decimal  by  the  square  of  the  radius  of  the  intrados;  ¥=1'"^ 

xC. 

Value 
of  the 

ratio 

«=4/ 

B-hf. 

8=%f. 

«=T/: 

«=s/: 

s=W. 

s^W. 

i,=58°^^80". 

29. 

r=f- 

5.S. 

r=W- 

r:^,2.!)f 

»=48°3640" 

«=86°52'10" 

r=81°58-26" 

«=28°420". 

B=:22''87'10' 

tJ^U"  15'. 
0.12760 

1.40 

0.16920 

0.16920 

0.16920 

0.16920 

0.16920 

0.15932 

1.39 

0.16463 

0.16462 

0.16463 

0.16463 

0.16463 

0.1.5490 

0.12397 

1.38 

0.16009 

0.16009 

0.16009 

0.16009 

0.16009 

0.1,5052 

0.12035 

1.37 

0.15558 

0.15558 

0.15558 

0.15558 

0.15558 

0.14617 

0.11677 

1.36 

0.15111 

0.15111 

0.15111 

0.15111 

0.15111 

0.14185 

0.11322 

1.35 

0.14666 

0.14666 

0.14666 

0.14666 

0.14666 

0.13756 

0.10969 

1.34 
1.33 

0.14225 

0.14225 

0.13787 

0 . 14225 

0.13787 

0.14225 

0.13787 

0.14225 

0.13787 

0.13330 

0.12908 

0.10619 

0.10271 

0.14138 

1.32 

0.14090 

0.13353 

0.13353 

0.13353 

0.13353 

0.12488 

0.09926 

1.31 
1.30 

0.14032 
0.13964 

0.12922 

0.12922 
0.12495 

0.12922 
0.12495 

0.12922 
0.12495 

0.12073 
0.11659 

0.09583 
0.09243 

0.12499 

1.29 

0.13885 

0.12425 

0.12071 

0.12071 

0.12071 

0.11250 

0.08906 

1.28 

0.13794 

0.12342 

0.11650 

0.11650 

0.11650 

0.10843 

0.08572 

1.27 

0.13693 

0.12250 

0.11232 

0.11232 

0.11232 

0.10439 

0.08240 

1.26 
1.25 

0.13579 
0.134.54 

0.12148 
0.12036 

0.10817 

0.10817 
0.10405 

0.10817 
0.10405 

0.10039 
0.09643 

0.07910 
0.07583 

0.10456 

1 .  24 

0.13316 

0.11914 

0.10359 

0.09997 

0.09997 

0.09249 

0.07259 

1.23 

0.13166 

0.11780 

0.10254 

0.09592 

0.09592 

0.08858 

0.06937 

1.22 

0.13002 

0.11635 

0.10138 

0.09190 

0.09190 

0.08469 

0.06618 

1.21 
1.20 

0.12824 
0.12632 

0.11478 
0.11309 

0.10012- 
0.09876 

0.08792 

0.08792 
0.08397 

0.08085 
0.07704 

0.06302 
0.05988 

0.08527 

1.19 

0.12426 

0.11127 

0.09728 

0.08412 

0.08005 

0.07325 

0.05677 

1.18 

0.12204 

0.10930 

0.09569 

0.08287 

0.07617 

0.06950 

0.05368 

1.17 

1.16 

0.11966 
0.11712 

0.10719 
0.10492 

0.09396 

0.09209 

0.08150 
0.08002 

0.07232 

0.06579 
0.06210 

0.05062 
0.04758 

0.06947 

1.15 

0.11440 

0.10248 

0.09007 

0.07840 

0.06819 

0.05845 

0.04457 

1.14 

0.11151 

0.09987 

0.08788 

0.07664 

0.06680 

0.05483 

0.04159 

1.13 
1.12 

0.10842 
0.10514 

0.09707 
0.09408 

0.08553 
0.08298 

0.07473 
0.07263 

0.06527 
0.06359 

0.05124 

0.03864 
0.03571 

0.04911 

1.11 

<». 10166 

0.09087 

0.08O22 

0.07034 

0.06173 

0.04791 

0.03281 

1.10 

0.09796 

0.08744 

0.07724 

0.06784 

0.05967 

0.04655 

0.02993 

1.09 

0.09403 

0.08376 

0.07401 

0.06510 

0.05738 

0.04.503 

0.02708 

1.08 
1.07 

0.08986 
0.08544 

0.07982 
0.07559 

0.07051 
0.06G71 

0.06209 
0.0537S 

0.05485 
0.05202 

0.04329 
0.04129 

0.02425 

0.02:51 

1.06 

0.08076 

0.07106 

0.06257 

0.05511 

0.04884 

0.03897 

0.02168 

1.05 

0.07579 

0.06620 

0.05806 

0.05106 

0.04526 

0.03629 

0.02064 

1.04 

0.07053 

0.06098 

0.05314 

0.04654 

0.04120 

0.03313 

0.01929 

1.03 

0.06495 

0.05536 

0.04775 

0.04149 

0.03656 

0.02935 

0.01756 

1.02 

0.05904 

0.04931 

0.04182 

0.03583 

0.03123 

0.02479 

0.01499 

|l.01 

0.05277 

0.04279 

0.03530 

0.02942 

0.02505 

0.01915 

0.01125 

320 


THEORY     OF    THE     ARCH. 


Table  F. — Circular  Arches  of  180°,  avith  a  load  of 
ox  each  side  of  the  central  ridge,  to  a  roof 
[/=tlie  angle  between  the  roof  aiul  a  vertical ;  ;'=:tlie 
extrados;  C=^ih.e  decimal  in  any  column;  jP=:  the 
springing  line.  The  last  two  columns  give  the  addition 
with    masonry,  and   of   the  uniform  depth  t  above  the 


1.50 
1.48 
1.46 

jl.44 

1.42 

1.40 

|1.39 
1.38 

1.37 
1.36 

1.35 

!l.34 
1.33 
1.32 
1.31 
1.3n 
1.2y 
1.28 
1.27 
1.26 
1.25 
1.24 
1.23 
1.22 
1.21 
1.2(1 
l.lH 
1.18 
1.17 
1.1(> 
1.15 
1.14 
1.13 
1.12 
1.11 
l.lo 
l.OK 
l.OH 
1.04 
1.02 


(2) 


4.000 
4.166 
4.347 
4.545 

4.761 

5.000 

5.128 
5.263 

5.406 
5.555 

5.714 

5.882 

6.060 

6.264 

6.451 

6.666 

6.896 

7.142 

7.407 

7.692 

8.000 

8.333 

8.695 

9.090 

9.523 

10.000 

10.526 

11.111 

11.764 

12.500 

13.333 

14.285 

15.384 

16.666 

18.181 

20.000 

25.000 

33.333 

.lO.OOO 

100.000 


7=90° 


O  3 


F=rt  X  C 
(8) 


1.003S2 


(4) 


0.21673  ••.'2( (535 
0.2i»6!t(in. 19588 
0.19733  u.  1865-1 
0.18782  0.17733 


0.17845 


0.16920 


0.16463 
0.16009 


0.15558 
0.15111 


0.14666 


0.16824 


0.15928 


0.15485 
0.15045 


0.14608 
0.14174 


0.14094 


0.14491 

0.14467 

0.14460 

0.1439(1 

0.14332  0. 

0.14264  0. 

0.14186'(). 

0.141iHiU. 

0.139880. 

0.13872'0. 

0. 13737 'o. 

0.13593!0. 

0.1343710. 

(J.132(i3  0. 

0.131173  0. 

0.12870,0. 

0.1265i»;0. 

0.12415  0. 

0.121820. 

0.11895(1. 

(».ii«(t«:o. 

O.Tl.3(i3|(). 
0.109791(1. 
0.lO(i41  (). 
0.10279JO. 
0. 09497 10. 
0.08638  (I. 
0.076H6  (I. 
0.06647|(). 


,14044 
,13984 
,13913 
,13830 
1.373(1 
13631 
13512 
13381 
13242 
13089 
12923 
12743 
12.550 
1 2339 
12116 
11877 
11623 
11352 
11063 
10759 
10443 
10105 
09749 
09373 
08978 
0814(» 
07213 

061KC. 

05112 


7=80° 

JFzrTfx 

1.0154.3 


(5) 


0.19S83 
'1.18952 
0.18O.33 
0.17127 

0.16234 

0.15353 

0.14917 
11.14484 


0.14221 
0.14157 


0.14083 


14002 
13908 
13805 
1.3(J91 
13567 
13431 
13282 
13121 
12948 
12762 
12563 
12350 
12124 
11883 
11628 
1 1 357 
11071 
1(J768 
10450 
10119 
09768 
09398 
09009 
08601 
08175 
07264 
06281 
O52o7 
(»4o34 


7=75* 

1.03528 


(6) 


0.19787 
0.18858 
0.17941 
0. 17030 

0.16144 

0.15265 


0.14840 
0.14768 


0.14685 
0.14593 


0.14492 


.14380 
. 14258 
.14126 
.13984 
.13832 
.13666 
.1349(( 
.13303 
.131(13 
.12891 
.12666 
.12427 
.12175 
.11909 
.11628 
.1133-i 
.1102O 
.10693 
.10349 
. 09989 
.09609 
.09212 
.08796 
.08360 
.07903 
.06925 
.05865 
.04707 
.((3422 


7=70° 
E=Ry. 

1.(16418 
F=r^C. 


(T) 


0.20289 
0.19346 
0.18416 
0.17498 

0.16594 


0.15877 

0.15783 
0.15688 

0.15583 
0.15468 

0.15344 


7=65° 
E=Rx 

1.11I.S.SS 
F=r^C. 


(8) 


0.21470 
0. 20498 
0.19539 
0.18594 

0.17661 


0.17307 

0.17201 
0.17086 

0.16962 
0.16829 

0.16686 


,1521(t 
,15066 
,14914 

,1475ii|0 
14.571  0 
14385  0 
14188  0 


13979 

13761 

13525 

13277 

13018 

12745 

124.58 

12155JO 

11840(1 

11515  () 

11166  0 


,10804 
,10426 
,  10028 
,09612 
,09178 
,08725 
,08250 


.16534 

.16372 

.16200 

.16(H7 

.15824 

.15620 

.  1 54(  (5 

.15178 

.14940 

.14693 

.14434 

.14163 

.13878 

.13578 

.13261 

.12938 

.126(12 

.12249 

.11876 

.11498 

.11106 

. 106K5 

.10249J46 

.09792  45 

.09312  44 


7=60° 
A'=7?x  1.15470 


•sB    7'=rs<7. 


(9) 


23"  0.234O8 
23  |0. 22388 
23  0.21381 
23    0.2O387 


58 


58 


58 


0.19510 


0.19291 


0.19170 
58  |0. 19041 


0.18903 
0.18755 

0.18599 

0.18433 
0.18257 
0.18070 
0.17877 
0.17674 
0.17458 
0.17232 
0.16996 
0.16750 
0.16492 
0.16224 
0.15946 
0.15654 
0.15353 
0.15038 
0.14713 
0.14375 
0.14027 
0.13669 
0.13294 
0.12913 
0.12521 
0.12119 
0.11710 
0.11290 


TABLES. 


321 


MASONRY  OR  OF  EQUAL  WEIGHT  WITH  MASONRY,  RISING 
TANGENT   TO    THE    EXTRADOS. 

radius  of  the  intrados ;  E=Kr—i\iQ  radins  of  the 
thrust;  .^=the  elevation  of  the  ridge  aljove  the 
to  the  thrust  caused  by  a  surcharge  of  equal  weight 
roof,  in  the  case  of  rotation  and  the  case  of  sliding.] 


7=55° 

1.2'207S 


(10) 


0.26225 
0.25133 

0.24056 


7=50° 
E=R^ 
1  30541 


(11) 


7=45° 
^'=7^x  1.41421 


■r,S    i?'=r-2(7. 


(12) 


0 . 30085 
0.28891 
0.27712 


0.22993  0.26551 


0.22178 

0.21941 

0.21810 
0.21671 

0.21522 
0.21365 

0.21199 

0.21023 
0.20838 
0.20643 
0.20437 
0.20227 
0 . 20009 
0.19780 
0. 19540 
0.19289 
0.19027 
0.18757 
0.18481 
0.18192 
0.17890 
0.17588 
0.17273 
0.16944 
0.16617 
0.16273 
0.15913 


0.25665 


0.25418 


0.25284 
0.25138 


22^ 


60 


59 


0.24985  59 
0.24826  59 


0.24653 


24473 
24284 
24086 
23876 
23670 
23451 
23220 
22983 
22732 
22478 
22219 
21948 
21665 
21385 
21095 
20790 
20493 
20182 
19857 
19515 


J. 35206 
1.33934 
1.32621 
0.31325 


Add  for  a  ]  Add  for  a 
surch'ec  ofisurchai'ge 

uniform    |    of  uni- 
depth  i,  the 


addition= 
A  =  rUl 
Rotation, 
(18) 


The  addition  to 
taken  from  this  c 
umn  when  the  llir 
omes  below  the  h( 
zontal    line   i.ear    i 


top  of  P 


0.30296 

0.30001 

0.29712 
0.29706 

0.29550 
0.29386 

0.29285 

0.29037 
0 . 28850 
0 . 28654 
0.28456 
0.28231 
0.28027 
0.27810 
0.27578 
0.27343 
0.27102 
0.26850 
0.26608 
0.26377 
0.26074 
0.^5806 
0.25546 
0.25277 
0.25010 
0.24742 
0.24477 
0.24218 
0.23967 
0.23732 
0.23502 
0.23292 


65° 

65 

64 
64 

64 
64 

63 

63 
63 
02 
62 
62 
01 
61 
60 
60 
60 
59 
59 
58 
58 
57 
50 
56 
55 
55 
54 
53 
52 
52 
51 
50 
47 
44 
41 
35 


form 

depth  t, 

A-rtC. 

Sliding. 

(14) 


0.33918 

0.35297 

0.3.5998 
0.36705 

0.37421 
0.38146 

0.38880 

0.39625 
0.40379 
0.41143 
0.41 920 
0.42711 
0.43513 
0.44329 
0.45161 
0.46009 
0.46875 
0.47760 
0.48665 
0.49592 
0.50543 
0.51520 
0.52527 
0.53564 
0.54637 
0.55748 
0.56901 
0.58102 
0.59359 
0.60676 
0.62063 
0.63532 
0.66778 
0.70588 
0.75313 
0.81867 


0.44388 
0.43796 
0.43204 
0.42612 

0.42020 

0.41429 

0.41133 

0.40837 

0.40541 
0.40245 

0.39949 


ne  near  th 
:ip  of  each 
column. 


fi^K 


The  fiddi 
tion  to  b> 
mtule    from 

lis  colu 

hen  the 
thrust  conies 
above      the 


ntnl 


Angle  oi 
rupture  25'^. 
which  Eivt' 
very  nearl; 
the  m^i'r 
eflfect  t^th 
surcharge. 


1.50 
1.48 
1.46 
1.44 

1.42 

1.40 

1.39 
1.38 

1.37 
1.36 

1.35 

1.34 

1 

1.321 

1.31[ 

1.30 

1.29 

1.28 

1.27 

1.26 

1.25 

1.24 

1.23 

1.22; 

1.21 

1 .  20 

1.19 

1.18 

1.17 

1.16 

1.15 

1.14 

1.13 


1.10 
1.08 
1.06 
1.04 
1.02 


322 


THEORY     OF    THE     ARCH. 


Table  G. 

Elliptical  Arches  of   180°,  avitii  a  load  of  masonry, 
OR  of  equal  weight  avitii  masonry,  rising  to  the 


level  of  the  top  of  the  key. 


[r=tlie  lialf  s]);!!! ;  f=:t]\e  rise  ;  C=^t\ie  decimal  in  any 
column;  i'^^tlie  tlirnst  =  /-61  The  true  thrust  due,  in 
every  case,  to  rotation.] 

TaMe  of  Thrusts  in  Eight  Systems. 


SJ3  g 

(1) 


~5 

the  span. 

(2) 


6.00 

6.50 

7.00 

7.50 

8.00 

8.50 

9.00 

9.50 
10.00 
10.50 
11.00 
12.00 
13.00 
14.00 
15.00'n. 
16.00 
17.00 
18.00 
19.00,0. 
20.00'(» 

2i.oo!o 

22.000 
23.00  0 
24.00y( 
25.0010 
26. 00  jo 
28.00l0 
30.00  0 
33.O0I0 
36.00(1 
40.00  0 

45.oo;o 

5O.OO1O 
55.00  0 
60.00  0 


22374 

21809 

21298 

20393 

19('.33J0 

18950  0 


^-r 


4 

the  span. 
(3) 


23222 

22387 
21684 
21004 
20513 


/- 


50 
27* 


100 
the  span. 

(4) 


18355 

17817 

17312 

16801 

10442 

10(i54;O, 

15091)  0, 

15355  (I 

15042'0, 

]474oIo 

1446910, 

14208  I), 

137300. 

1330810 

12740  0 

l'V25<i'o 

11 087  [0 

11110  0 

106280 

102220 

09874  0 


200(15 

19504 

19143 

18756 

18050 

17420 

10855 

10349!() 

15881  0 

15449  0, 

15000 

14090 

14359 

14040 

13753 

134SO 

132310, 

13005  <». 

12707 1(», 

12347  0, 

11977  ;0, 

11491  0 

11070'0 

loooolo 

10108  0 
097030 
09392  0 
0907510 


f=, 


10 
the  span. 


(5) 


23337 

22332 

21598 

20932 

20347 

19824 

19351 

18915 

18511 

18134 

17780 

17133 

10555 

10"29 

15552 1 0 

151100 

1471N  0 

14348 

14008 

13092 

1339S|o 

13131 

12SK5 

1 2035 

12407 

12197 

11810 

11463 

nolo 

10625 
10185 
09728 
09383 
09065 
08779 


,21612 

,  20903 

,20277 

,19725 

,19221 

,18779 

,18359 

,17971 

,17604 

,17283 

,16939 

,16340 

,15798 

,15300 

.14803 

.14455 

.14070 

.13730 

.13410 

.13120 

.12854 

.12584|o 

.1234110 

.12118|o 

.  11900|<l 

.1170810 

.11340  0 

.110200 


/=- 


the  span. :  the  span. 
(6)     (T) 


10004 
10240 
09827 
09428 
09089 
08770 
08520 


.19848  0 
.19312  0 
.18830  0 
.18390  0 
.1797710 
.17594]0 
. 17231 10 

.10904  jo 

.10504,0 
.I6255IO 
.15909J0 
.I5422I0 
.1492010 
. 14477 jO 
.14005  0 
.13687  jo 
.13341  0 
.13027  0 
.I2740I0 
.12453;0 
.  12200  (t 
.  1195".tO 
.11737  o 
.11521' 0 
.11330  0 
.111.54|0 
.10823  0 
.10527jO 
,10139JO 
.Op«04  0 
.09118,0 
.090690 
.08727  0 
.08455  0 
.08238  0 


fAr 


5 
the  span. 

(S) 


.18082 
.17714 
.17358 
.  17014|0 
.16080  0 
.1030910 

.  10049  lo 

.15755V) 
.15474|0 
.151990 
.14936  0 
.14452  0 
.14006:0 
.13599!o 
.13225  0 
.12889:0 
.12581  jo 
.12278  0 
.12009  0 
.11758  0 
.115270 
.1131010 
.  IIIO9J0 
.10924:0 
.lo747!o 

.  10581  ;o 

.10278  0 
.  10011  0 
.09659  0 
.0038'i  0 
.09047:0 
.08673'0 
.0838610 
.08148  0 
.0794310 


20 
the  span. 


(!>) 


17190 

16904 

16605 

16304 

16015 

1571010 

1543o;o 

1510-1  () 

14901 '0 

14643  0 

14401  0 

1394o'o 

13520 

13141 

12795 


12476 


12167 

11892 

11636 

11399 

11180 

10977 

10789 

looo'.)|o 

1044 -lO 


10280 

10001 

09745 

09431 

09172 

08823J0 

084H4  0 


08217 
07987 
07796 


.15706 
.15523 
.15330 
.15081 
.14846 
.14606 
.14367 
.14132 
.13907 
.13686 
.13451 
.13066 
.12095 
.12368 
.12035 
.11745 
.11472 
.11226 
.10997 
.10788 
.10589 
.10405 
.10233 
.10073 
.09924 
.09781 
.09528 
.09321 
.09031 
.08755 
. 08400 
.08169 
.07922 
.07720 
.07550 


TABLES. 


323 


Table  H. 

Thrust  of  the  Unloaded  Elliptical  King  bounded  by 
SIMILAR  Ellipses. 

[T'lrrtLe  span  ;  y=tlie  rise  ;  c/mtlie  thickness  at  the  key ; 
(7=  the  decimal  in  any  column;  i^=the  thrust =7-^ 6'/ 
semi-axes    of   the   intrados,  f  and  r  /    semi-axes    of  the 

V 

extrados /+<;/,  and  r-\--,d7\ 

Thrust  in  Two  Sy-s-tems. 


Value  of 

Pase=i  the 
span. 

1 
Kise=j  the 

O 

span. 

Value  of 

Kise=-  the 
-span. 

Eise  =  -  the 
o 

span. 

2r 

/^r- 

/4- 

2r 
~d 

^=r- 

f^'- 

F=r-^^ 

F=r-^>i 

F=r-^  X 

Fy.r'^y. 

6.00 

0.18273 

19.00 

0.11725 

0.(19559 

6 .  50 

0.17772 

20.00 

0.11305 

0.09222 

7.00 

0.19773 

0.17255 

21.00 

0.10953 

0.08904 

7.50 

0.19412 

'    0.16761 

22.00 

0.10614 

0.08592 

8.00 

0.18893 

0.16265 

23.00 

0.10291 

0.08304 

8 .  50 

0.18431 

0.15782 

24.00 

0.09981 

0.08050 

9.00 

0.17970 

0.15341 

25.00 

0.09685 

0.07815 

9.50 

0.17544 

0.14938 

26.00 

0.09419 

0.07575 

10.00 

0.17124 

0.14621 

28.00 

0.08924 

0.07144 

10.50 

0.16776 

0.14132 

30.00 

0.08468 

0.06770 

11.00 

0.16310 

0.13730 

33.00 

0.07889 

0.06241 

12.00 

0.15574 

0.13030 

36.00 

0.07364 

0.05835 

13.00 

0.14968 

0.12376 

40.00 

0.06779 

0.05364 

14.00 

0.14233 

0.11799 

45 .  00 

0.06137 

0.04867 

15.00 

0.13686 

0.11242 

50 .  00 

0.05663 

0.04459 

16.00 

0.13142 

0.10783 

55.00 

0.05235 

0. 041 7 2 

17.00 

0.12621 

0.10352 

60.00 

0.04870 

0.03815 

18.00 

0.12174 

0.09941 

324  THEORY    OF    THE    ARCU. 


SECTION  VI. 


CURVE    OF    PRESSURE. 


108.  In  tlie  preceding  part  of  tliis  work,  we  have 
followed  the  theory  of  Coulomb,  Aiidoy,  and  Poncelet, 
and  calculated  the  thrust  of  arches  at  the  moment  of 
rupture.  We  have  called  this  the  maximum  thrust  of  the 
arch. 

At  that  imaginary  moment,  the  horizontal  thrust  at 
the  key,  in  the  ordinary  mode  of  rupture,  acts  upon  a 
single  point  or  line  of  the  extrados ;  the  resultant  of  this 
thrust,  and  of  the  weight  of  that  part  of  the  arch  and  its 
load  which  lies  above  the  joint  of  rupture,  comes  upon  the 
intrados  of  that  joint ;  and  the  resultant  of  the  same 
thrust,  and  of  the  weight  of  the  whole  semi-arch  and  pier, 
falls  entirely  upon  the  exterior  edge  of  the  l)ase  of  the 
pier  (see  figures  2,  3). 

No  masonry  could  withstand  this  pressure  exerted 
upon  mere  points  or  edges. 

Not  only  must  rupture  he  avoided,  but  we  must  take 
care  not.  to  approach  that  condition ;  that  is,  we  must,  if 
possible,  so  pro])(>rtion  and  build  the  arch  and  its  pier,  as 
to  keep  the  curve  of  i)ressure  near  the  middle  of  the 
joints. 


CURVE     OF     PRESSURE. 


Let  us  suppose  any  arcli  to 
be  on  the  point  of  falling  in 
consequence  of  the  insufficient 
thickness  of  its  pier.'  The 
curve  of  pressure,  touching  the 
extraclos  at  the  key,  and  the 
intrados  at  the  reins  about  60° 
from  the  key,  passes  finally 
through  the  exterior  edge  of 
the  base. 

Before  reaching  this  con- 
dition of  rupture,  the  vertical 
joint  has  gradually  opened  on 
the  lower  side,  the  joint  at  the  reins  on  the  exterior 
side,  the  pier  has  slightly  yielded  to  the  pressure  of  the 
arch,  the  key  has  settled  down,  the  reins  have  spread  out. 

These  movements,  from  which  the  best  of  arches  are  not 
entirely  free,  are  often  developed,  in  badly  proportioned 
works,  so  as  to  exhibit  wide  cracks  at  the  key  and  reins, 
without  any  immediate  danger  to  the  structure.  They 
are  due  to  two  principal  causes:  in  single  arches,  to 
deficiency  of  mass  in  the  pier ;  in  continuous  arches,  to  the 
absence  of  suitable  arrangements  for  preventing  lateral 
motion  at  the  reins. 


FiQ.  23. 


109.  In  single  or  ahdment  arclies,  tlie  magnitude  of 
tlie  horizontal  thrust,  and  the  place  of  the  curve  of  pressure 
in  the  arch,  depend  largely  upon  the  dimensions  of  the  pier. 

Let  us  for  a  moment  admit  that 
the  pier  or  abutment  is  absolutely 
immovable,  and  that  the  material  of 
the  arch  is  susceptible  of  but  very 
little  compression.  And  let  us  fur- 
ther admit,  what  has  been  abund- 
antly proved  in  this  paper,  and  con- 
firmed by  numerous  observations,  that 
the  joint  of  rupture,  about  G0°  from 


Fig.  -24. 


32  G  THEORY     OF    TQE    ARCH. 

tlie  key,  is  the  weakest  joint,  or  joint  first  to  open  at 
the  extrados.  As  the  arch  l)eh:>\v  that  joint,  ;?r??,  being 
firmly  attached  to  the  pier,  is  immovable,  and  the  masonry 
above  that  joint,  by  supposition,  nearly  incompressil^le,  it 
is  evident  that  the  pressure  at  the  key,  a  h^  and  at  the 
joint  of  rvidure^  m  n^  will  act  all  along  those  joints ;  in 
other  words,  those  joints  will  be  everywhere  in  contact. 
Ill  the  most  perfect  condition  of  stability,  the  resultant 
of  the  horizontal  forces  acting  along  the  key,  a  b,  will  pass 
through  the  middle  of  that  joint;  in  like  manner  the 
resultant  of  all  the  forces  acting  along  m  ??,  normal  to  that 
joint,  will  pass  through  its  middle  point. 

Let  us  now  suppose  the  thickness  of  the  pier  to  be 
gradually  diminished  until  its  top  begins  to  move  away 
from  the  arch  :  the  crown  will  l)egin  to  settle,  the  i-eins 
will  spread  out,  the  curve  of  pressure  will  approach  the 
extrados  at  the  key  and  the  intrados  at  the  reins ;  finally, 
when  the  pier  has  been  sufiSciently  reduced,  the  curve  of 
pressure  will  pass  through  the  extremities  of  those  joints. 

Thus,  by  mere  external  changes  in  the  pier,  we  have 
caused  the  curve  of  pressure  in  the  arch  to  move  by  de- 
grees, from  the  place  of  most  perfect  stability,  to  that  of 
final  rupture  and  fall. 

In  this  final  condition,  the  thrust  of  the  arch  is  the 
horizontal  force  which,  applied  at  the  extrados  of  the  key, 
is  just  suflficieiit  to  prevent  the  rotation  of  the  segment 
above  the  joint  of  rupture  around  the  intrados  of  that 
joint ;  this  force  acting  with  a  lever  arm  equal  to  the  ver- 
tical distance  between  these  points.  On  the  other  hand, 
in  the  condition  of  most  perfect  stability,  the  acting  thrust 
is  the  horizontal  force  which,  applied  to  the  middle  of  the 
key,  is  just  sufiicient  to  prevent  the  rotation  of  the  same 
upper  segment  around  the  middle  of  the  joint  of  rupture  ; 
this  force  acting  with  a  lever-arm  equal  to  the  vertical 
distance  between  these  middle  points. 

This  real  thrust,  acting  when  the  arch  is  firmly  estab- 


CURVE    OF    PRESSURE.  327 

lislied  upon  its  piers,  is  miicli  greater  than  the  final  thrust 
exerted  at  the  moment  of  rupture, — sometimes  more  tlian 
double. 

We  thus  perceive  that  what  has  been  called  the  maxi- 
mum thrust  in  the  former  part  of  this  work,  is  really  tlu^ 
least  thrust  that  can  ever  act  at  the  crown  of  the  arch, 
and  that  this  minimum  is  attained  at  the  moment  of  rup- 
ture. 

The  effective  thrust  is  increased  in  the  same  arch  as  its 
lever-arm  is  diminished,  or  as  the  curve  of  pressure  falls  at 
the  key  and  rises  at  the  reins. 

In  ordinary  circular,  segmental,  and  elliptical  arches, 
surcharged  horizontally,  or  surcharged  more  at  the  key 
than  at  the  reins,  the  curve  of  pressure  can  never  fall 
below  the  middle  of  the  key,  or  rise  above  the  middle  of 
the  reins. 

110.  Tlie  effective  liorizontal  thrust^  the  place  of  tlie 
curve  of  pressure^  and  the  stahility  of  contimiou-s  arclies^ 
resting  on  intermediate  piers^  depend  largely  tipon  the 
material  of  the  top  of  the  pier  hetiveen  tJte  arches.  If  this 
filling  be  of  earth,  or  of  indifferent  masonry,  the  reins  of 
the  arch  will  spread  out,  the  curve  of  pressure  Avill  rise  at 
the  crown  and  fall  at  the  reins,  the  key  will  settle  down, 
cracks  make  their  appearance,  and  the  whole  assume  an 
appearance  of  instability,  or  even  worse. 

The  remedy  of  this  is  simple  and  certain. 


Fig.  2.5. 

The  arch,  if  li2:ht,  should  increase  in  thickness  from  the 

key  towards  the  springing  line,  so  as  to  add,  60°  degrees 

from  the  ke}^,  at  least  fifty  per  cent,  to  the  thickness  at 

the  key ;  and  the  spaces  between  the   arches,  over  the 

10 


3-JS 


TUEORY    OF    THE    ARCH. 


piei-s,  must  be  filled  witli  closely  jointed,  solid  masonry,  in 
liorizontal  courses,  abutting,  in  vertical  joints,  upon  the 
adjacent  voussoirs,  and  extending  as  liigh  as,  say,  within 
45°  of  the  crown.  These  precautions  are  best  illustrated 
in  the  London  Bridsce,  the  most  remarkable  structure  of 
its  kind,  perhaps,  in  the  world.  A  reversed  arch,  of  equal 
thickness  Avith  the  arch  proper,  is  laid  npon  the  top  of 
each  pier,  abutthig,  in  vertical  joints,  upon  the  voussoirs 
of  the  reins  and  lower  j^arts  of  the  adjacent  arches.  The 
joints  are  as  thin  as  possible ;  and  no  other  motion  can 
occur  in  the  arch  than  the  little  which  arises  from  the 
compressibility  of  the  granite.  These  precautions  secure 
to  tlie  semicircular  or  elliptical  arch,  all  the  stiffness  and 
stability  of  the  segmental  arch. 

111.  The  curve  of  pressure  at  any 
joint  should  7iot pass  within  one  third 
of  its  leyigth  from  either  edge. 

Suppose  the  pressure  to  be  nothing 
at  the  intrados,  a^  and  to  increase  uni- 
formly from  that  point  to  the  extra- 
dos,  h.  It  is  plain  that  the  pressure  at  any  point  along  a  h^ 
will  be  represented  by  the  ordinate  of  a  certain  triangle. 
Tlie  whole  pressure  will  be  represented  by  the  surface  of 
that  triangle;  and  the  point  of  application  of  the  resultant 
of  all  the  pressures  Avill  Ije  at  c,  opposite  the  center  of 
gravity  of  that  triangle.  We  then  have  cZ*— Jrt  ^.  Vice 
versa,  if  tlie  point  of  aj^plication  be  at  c,  <?^=:^  rtZ*,  we 
know  that  tlie  pressure  is  nothing  at  a. 

If  the  point  of  api)lication  be  at  c^  ch  being  less  than 
J  <(  />,  G  being  still  opposite  tlie  center  of  gravity  of  tlie 
triangle  whose  ordinates  represent  the 
]>res>ure,  we  know  that  the  vertex  of 
that  triangle,  and  point  of  no  i:>res- 
sure,  are  at  <?,  he=z2>xh  c. 

In  this  case,  the  joint  a  h  will  open 
at  rt,  as  far  as  e ;  the  adjacent  joints 


Fio.  26. 


Fio.  27 


TRESSURE     PER     UNIT     OF    SURFACE. 


329 


will  also  open  until  we  come  to  one  where  the  curve  of 
pressure  passes  within  the  prescribed  limit. 

This  reasoning  is,  of  course,  applical)le  to  all  the  joints  ; 
and  Ave  readily  conclude  that  the  curve  of  pressure  should 
lie  entirely  between  two  other  curves  which  divide  the 
joints  into  three  equal  parts. 

The  foregoing  reasoning  is  based  on  the  principle,  first 
applied,  we  believe,  by  NaAder,  that  the  material  of  the 
arch  is  perfectly  elastic,  and  that  the  pressure  upon  any 
joint  varies  uniformly  from  the  extremity  most  pressed  to 
the  point  of  no  pressure.  In  fact,  the  last  condition  alone 
is  sufficient,  as  it  allows  us  to  represent  the  pressure,  upon 
the  several  points  of  any  joint,  by  the  ordinates  of  a 
triangle  or  trapezoid. 


PRESSURE,    PER   UJSTIT    OF    SURFACE,    UPON   THE   JOINTS. 

112.  Let  i^  represent  the  total  perpendicular  pressure 
upon  any  joint ;  d^  the  length  of  the  joint ;  Z,  the  distance 
between  the  resultant  or  curve  of  pressure,  and  the  nearest 
edge  of  that  joint;  P,  the  pressure  per  unit  of  surface  at 
the  edge  most  exposed. 

If  the  curve  of  pressure  pass  through  the  middle  of  the 

jonit,  we  nave  ±^=.-7. 

If  the  curve  of  pressure  pass  at 

one  third  the  length  of  the  joint  from 

either  edge,  as  Z*,  we  have  the  pressure 

at  h  equal  to  twice  the  mean  pressure 

2i^ 
along  a  ^,  or  P= — j-. 

If  I  be  less  than  J  J,  or  c  I  less  than 
J  a  l^  the  whole  pressure  comes  upon 

e  h.  and  we  have  P^z"^^-. 

We  have  no  good  means  of  esti- 
mating the  distance  eh ;  but,  what  is 


Fig.  27. 


330 


THEORY    OF    THE    ARCH. 


more  to  the  purpose,  we  can  generally  confine  the  curve 
of  pressure  within  the  two  curves  above  mentioned,  or 
even  keep  it  still  nearer  the  middle  points  of  the  joints. 

The  pressure  being  nowhere 
nothing  ui)on  the  joint  itself,  will 
be  represented  by  the  ordinates  of 
some  trapezoid  ah  F F\  F^  F\ 
representing  the  pressures,  per  unit 
of  surface,  at  h  and  «,  respectively, 
we  shall  have,  F  :  F' ::b  F:a  F' ; 

F+F^ 
moreover,  F=^—^ — X</. 

The  resultant  of  all  the  pres- 
sures will  pass  through  the  center 
of  gravity  of  the  trapezoid. 

Let^),^>',  c,  1)6  the  projections,  upon  ab,  of  the  centers 
of  gravity  of  the  two  triangles  a  IF,  a  F  F\  and  the 
whole  trapezoid  a  h  F  F\  respectively.  The  point  c  i^ 
found  by  dividing^;'  j9  in  the  invei-se  ratio  of  «  P'  to  b  F. 
"VVe  have 

f  c'.cp-.'.lF'.aF'  ::P:  P' ; 

F  :  P+P"  ;  tciving  F=^'^^^  : 


Fic.  2S. 


p'  C  \ 2^^ p 


2F 


IhP 


buti/  c=e'/-/;  P+P'^:--^;  p'r  =  i'l 


Hence  Pz=— y-x^-.— 
a  ^a 

is  the  mean  pressure  per  unit  of  surface. 


(59) 


p. 

d 

From  (50)  the  values  of  P  already  given,  are  readily 

deduced.     The  formula,  however,  is  not  ap})licable  to  the 

case  in  which  the  point  of  no  pressure  comes  within  the 

extremities  of  the  joint. 


ACTUAL    THRUST. 


331 


THE  TRUE  THRUST  OF  THE  ARCH. 

113.  We  liave  reminded  the  reader  tliat  almost  all 
arclies,  on  the  removal  of  the  center,  show  a  tendency  to 
settle  down  at  the  key  and  spread  out  at  the  reins.     This 

b 


Fig.  29. 


tendency  often  results  in  the  production  of  cracks,  at  the 
intrados  of  the  key,  and  at  the  extrados  of  the  reins  on 
each  side  of  the  key. 

Suppose  the  crack  at  the  key  to  extend  from  a  to  a 
certain  points;  the  whole  pressure  comes  upon  eh^  the 
remaining  part  of  the  joint,  and  all  that  part  of  the  arch 
near  the  key  which  lies  below  the  joint  e,  is  worse  than 
useless  ;  in  like  manner,  supj)osing  the  joint  at  the  reins  to 
open  from  n  to  e\  that  part  of  the  arch  which  is  external 
to  6'  is  useless.  If  mere  weight  be  wanted  at  any  point,  it 
will  be  better  to  load  the  arch  with  some  cheap  material. 

We  have  also  reminded  the  reader  of  the  self-evident 
truth,  that  the  movements  and  cracks  in  question  will  always 
be  developed  unless  the  pier  opposes  a  sufficient  resistance. 
If  the  pier,  though  large  enough  to  prevent  actual  rupture 
and  fall,  is  still  weak,  the  crown  will  continue  to  settle, 
and  tlte  Jiorizontal  thrust  to  diminisli.,  until  the  pier  is  able 
to  withstand  the  diminished  thrust.  The  arch,  in  dimin- 
ishing its  thrust,  tries,  as  it  were,  to  accommodate  itself  to 
the  weakness  of  the  pier.  Between  the  condition  of  most 
perfect  stability,  tlie  curve  of  pressure  passing  near  the 
middle  of  the  joints,  and  the  condition  of  final  rupture  and 
fall,  the   existing  thrust  becomes   less  and  less,  varying 


332 


TUEORY     OF    THE    ARCH. 


sometimes,  as  avc  shall  liereafter  see,  iu  a  ratio  as  great  as 
2  to  1,  or  larger  still. 

This  condition  of  most  perfect  stability  is  highly  favor- 
able to  the  joints  of  the  arch,  the  pressure  being  nearly 
equally  distributed  ;  but  it  is  the  condition  which  gives 
rise  to  the  greatest  thrust,  and  requires  the  greatest  mag- 
nitude of  pier. 

We  can  not  say  that  the  pier  ought,  in  all  cases  to  be 
large  enough  to  withstand  so  great  a  thrust ;  but  it  is  very 
certain  that  the'  pier  ought  to  withstand  that  diminished 
thrust  which  is  developed  when  the  curve  of  pressure  at 
the  crown  and  at  the  reins,  passes  through  the  limits 
already  fixed ;  viz.,  at  the  key,  ^  the  length  of  the  joint 
from  the  extrados,  and  at  the  reins,  -^  the  length  of  the  joint 
from  the  intrados.  If  the  pier  cannot  withstand  this 
thrust,  the  joints  of  the  arch  will  certainly  open. 

We  have  here  a  perfectly  distinct  point  of  departure 
for  a  new  calculation  of  the  thrust  of  arches. 

Draw  the  curves  a'  m\  V 
7i',  dividing  all  the  joints  into 
three  equal  parts.  Suppose 
the  horizontal  thrust  to  be 
applied  at  V  on  the  key,  V  h 

Draw  any  joint  m  m  n  n, 
the  vertical  7i  r  through  the 
surcharge,  and  the  line  r  t  re- 
presenting the  top  of  the  sur- 
charge ;  i)r()ject  m  horizontally  at  x  on  the  vertical,  Ch^ 
which  divides  the  arch  into  two  equal  parts ;  and  project 
^,  the  center  of  gravity  of  the  segment,  on  n  r  t  a  on  m'  x 
at  (J.  Let  y\z=zh' x^  represent  the  lever  arm  of  the  thrust, 
andy>',  =  m' ^',  the  lever  arm  of  the  segment. 

A'^  the  resultant  of  the  thrust,  applied  at  h\  and  the 
weight  of  the  segment  resting  on  tn  //,  aj)plied  at  ^,  its 
center  of  gi'avity,  passes,  by  supj^osition,  through  ni\  in  iii=i 


Fig.  80. 


ACTUAL    THRUST.  333 


^771 72,  there  must  be,  in  case  of  equilibrium,  an  equality  of 
moments  in  relation  to  that  point.  Hence,  ^'  represent- 
ing the  force,  and  S'  the  surface  m  nr  tha m,  we  must 
have 

y 

This  force  F'  will  be  small  when  the  joint  m  n  is  near 
the  key ;  it  will  increase  as  the  joint  departs  from  the 
key,  and  become  a  maximum  at  the  reins,  about  60°  from 
the  key. 

Suppose  m  7?,  to  be  the  joint  of  rupture  corresponding 
to  the  maximum  value  of  F'.  The  curve  of  pressure 
between  the  key  and  the  joint  of  rupture  will  be  situated 
entirely  between  the  limit  curves,  a'  m\  h'  n\  which  divide 
the  joints  into  three  equal  parts.  From  h\  where  it  is 
nearest  to  the  extrados,  it  will  gradually  depart  from  the 
extrados,  pass  through  the  middle  of  some  joint  about 
midway  between  a  h  and  m  7Z,  continue  to  approach  the 
intrados,  and  come  nearest,  relatively,  to  that  curve  at  m\ 
7nm=^m7i;  it  will  there  be  tangent  to  the  inferior 
curve  a'  m',  and  begin  to  recede  from  the  intrados.  In 
light  arches,  the  curve  of  pressure  after  leaving  b\  may  at 
first  pass  a  little  above  the  superior  limit,  h'  71  ;  but  it 
never  can  pass  within  the  inferior  limit,  a  on'^  either  above 
or  below  the  joint  of  maximum  thrust.  The  reader,  by  a 
little  reflection,  will  see  the  truth  of  this  last  remark,  and 
will  also  see  its  importance. 


334 


THEORY    OF    TUE    ARCn. 


THRUST  OF  THE  UNLOADED  CIRCULAR  KING  OF  EQUAL  THICK- 
NESS THROUGHOUT,  ON  THE  SUPPOSITION  THAT  NO  JOINT 
SHALL  OPEIT,  THE  CURVE  OF  PRESSURE  AT  THE  KEY  BE- 
ING   ONE    THIRD    THE     LENGTH    OF   THE   JOINT    FROM    THE 

EXTRADOS, AT   THE    REINS,  ONE   THIRD    THE   LENGTH    OF 

THE   JOINT   FROil    THE   INTRADOS. 


Fio.  81. 


114.  The  point  of  ap- 
plication of  the  thrust  is  at 
y ^  V  h^=^\a  h.  The  arcs,  a 
m\  h'  n\  diWding  the  joints 
into  three  equal  parts,  the 
segment  m  n  h  a^  corre- 
sponding to  any  joint  7n  ??, 
is  exactly  equal  to  three 
times  the  central  segment, 
m'  n  h'  a  /  and  the  center 
of  gravity  of  the  whole  is 
near  the  center  of  gravity 
of  this  central  part.  In  the  light  arches  of  ordinary  use 
these  centers  may  Le  regarded  as  coinciding.  Su2')pose 
them  to  coincide. 

The  thrust  of  the  ai'ch  vi  n  l>  a^  on  the  condition  ex- 
pressed at  the  head  of  this  article,  is  precisely  equal  to 
tliree  times  the  thrust  of  the  arch  ///'  n'  I'  a\  calculated  on 
the  condition  of  actual  rupture. 

This  last  thrust  for  all  prt)poi-tions  of  the  two  radii  is 
given  by  table  A,  from  which  the  following  has  been  de- 
duced.    We  briefly  explain  the  mode  of  conq^utation. 

Let  TT  represent  the  ratio  of  the  two  radii  of  the  given 
arch  ;  Jl'  the  I'atio  of  the  radii  of  the  central  arch  ;  ^' 
the  thrust  of  this  last  arch  taken  from  table  A  ;  I^^  the 
I'cquired  thrust  of  the  given  arch.  Vie  have 


TABLE     AA.  335 

Table  AA,  at  tlie  end  of  tins  paper,  gives  the  values  of 
F^  or  the  actual  tln'ust  of  the  ch'cular  ring,  for  all  the  val- 
ues of  ^between  1.01  and  1.40  inclusive. 

For  explanation  see  the  head  of  that  table. 

This  table  proves  that  the  effective  thrust  acting  when 
the  arch  is  firmly  established  upon  its  piers,  is  much 
greater  than  the  thrust  at  the  moment  of  actual  rupture. 
The  ratio,  (§,  of  these  two  thrusts,  beginning  at  1.065  for 
7^=1.01,  becomes  1.204  for  7^=1.08,1.35  for  ^=1.17, 
1.50  for  ^=1.25,  1.83  for  7^=1.40.  It  attains  its  great- 
est value,  1.94,  when  ^=1.45. 

The  values  of  7"  corresponding  to  7^=1.01,  1.02,  have 
been  calculated  by  the  law  of  differences,  table  A  not 
giving  those  values. 


THRUST    OF   SEMICIRCULAR    ARCHES    SURCHARGED    HORIZON- 
TALLY. 

115.  This  is  by  far  the  most  common  form  of  the  arch. 
All  arches  carry  loads,  and  these  loads  most  frequently 
rise  to  a  surface  nearly  horizontal. 

It  is  plain  that  the  pier  should  oppose  to  the  thrust  of 
the  arch  a  resistance  sufficient  to  prevent  the  formation  of 
cracks  or  openings  at  the  weakest  joints,  or  joints  which 
actually  open  in  case  of  rupture  and  fall.     Figs.  2,  3. 

The  actual  thrust  of  the  arch,  when  the  joint  at  the 
key  is  about  to  open  at  the  intrados,  and  the  joint  at  the 
reins  is  about  to  open  at  the  exti-ados,  is  evidently  the  very 
minimum  which  the  pier  is  required  to  oppose.  If  the 
pier  is  unable,  in  the  slightest  degree,  to  meet  this  thrust, 
it  is  evident  that  the  joints  in  question  will  open,  and  con- 
tinue to  open,  nntil  the  pier  is  able  to  withstand  the  dim- 
inished thrust,  or  until  the  structure  falls. 

Let  us  assume  that  the  pier  is  able  to  withstand  that 
greater  thrust  which  is  developed  in  the  condition  of  most 


336  THEORY     OF    THE    ARCH. 

perfect  stability,  wlieu  tlie  curve  of  pressure,  or  resultant 
of  all  the  pressures,  passes  tlirougli  the  middle  of  the  key 
and  the  middle  of  the  reins.  By  the  latter  we  mean  the 
weakest  joint,  generally  about  G0°  from  the  key. 

In  the  investigation  of  this  thrust  we  shall  suppose  the 
thickness  of  the  arch  to  be  the  same  throughout;  while 
in  practice,  as  we  have  repeatedly  stated,  this  thickness 
should  gradually  increase  from  the  key,  so  as  to  become 
about  fifty  per  cent,  larger  at  the  reins.  This  increase 
will  slightly  diminish  the  thrust,  and  slightly  elevate  the 
curve  of  pressure  at  the  reins.  The  curve  of  pressure 
passing  through  the  middle  of  the  joint  at  the  reins,  sup- 
posed to  be  equal  in  length  with  the  joint  at  the  key,  will 
pass  through  the  inferior  limit  of  the  former  joint  when 
increased  by  fifty  per  cent. ;  viz.,  within  one  third  of  its 
lenoth  from  the  intrados.  It  will  be  a  little  below  the 
superior  limit  at  the  key ;  tliat  is,  by  the  difference  be- 
tween I  and  -L,  or  by  \  the  depth  of  that  joint. 

Almost  all  large  bridges  are  exceedingly  light  in  their 
proportions  ;  they  are  made  generally  of  the  most  incom- 
pressible stone ;  and  it  is  not  too  much  to  say  that  their 
piers  should  be  able  to  withstand  the  thrust  in  question 
developed  in  the  condition  of  most  perfect  stability.  Ap- 
plied to  arches  very  heavy  in  their  proportions,  whether 
large  or  small  in  their  actual  dimensions,  this  would  per- 
haps be  an  exaggerated  thrust.  Such  arches  have  an 
excess  of  thickness  throughout,  and  require  no  increase  at 
the  reins. 

Assume  the  curve  of  pressure  to  pass  through  ^,  the 
middle  of  the  key,  and  to  touch  the  line  drawn  through 
the  middle  point  of  all  the  joints,  supposed  to  be  equal  in 
length,  at  c,,  on  some  joint,  m  ??,  such  as  to  give  the  great- 
est possible  thrust  on  the  condition  imposed. 

To  illustrate  this  method  in  advance  of  more  particular 
calculations,  suppose  the  thickness  of  the  arch  to  be  -jV 


ACTUAL    THRUST. 


tlie  span,  or  IC=1,10.  Let  us 
find  the  value  of  the  hori- 
zontal force,  F\  which,  ap- 
plied at  0,  the  middle  of  a  h, 
shall  hold  in  equilibrium  any 
segment,  a  m  n  7'  h  a.,  on  (?i, 
the  middle  of  the  lower  joint. 
Let  v=t\ie  angle,  m  C  a,  be- 
tween the  joint  m  n  and  a 
vertical ;  r=the  radius  of  the 
intrados  ;  m  7i=a  b  /  in  all 
c'dBesjnc^=^m7i=BB:j  ^-m^i,, 
the  joint  of  the  practical  arch. 


t:,   f. 


Fig.  32. 


, 

Tallies  of  ■». 

Values  of  J^^ 

Values  of  v. 

Values  of  F'.                       j 

0° 

r'  X  0.10492 

50° 

r^x  0.13563 

5 

"  0.10544 

55 

"  0.13746 

10 

"  0.10698 

58 

"  0.13795 

15 

"  0.10945 

59 

"  0.13801  =i^,  the  max'm. 

20 

"  0.11270 

60 

"  0.13800 

25 

"  0.11654 

65 

"  0.13708 

30 

"  0.12073 

70 

"  0.13457 

35 

"  0.12504 

.75 

"  0.13038 

40 

"  0.12914 

80 

45 

1 

"  0.13277 

90 

"  0.10825 

The  angle  of  maximum  thrust  is  in  this  case  59  or  60 
degrees,  and  the  curve  of  pressure  corresponding  to  that 
maximum,  passing  through  the  middle  of  the  key  and  the 
middle  of  the  reins,  59°  from  the  key,  is  traced,  necessa- 
rily, outside  or  above  the  central  point  of  every  other  joint. 
If  it  were  "inside  of  the  middle  of  any  joint,  as  at  t'=20°,  it 
would  immmediately  follow  that  the  value  of  F'  corre- 
sponding to  the  middle  of  that  joint,  must  be  greater  than 
F^  the  actual  maximum  thrust. 

We  may  learn  from  the  above  table  that  the  curve  of 
pressure  corresponding  to  the  maximum  F^  passing  through 
the  middle  of  the  key  and  of  the  reins,  is,  everywhere  be- 


338  THEORY     OF    THE     ARCH. 

tween  those  joints,  very  near  the  central  points,  since  the 
maximum,  F^  so  little  exceeds  the  other  values  of  F' . 

In  the  uj^per  parts  of  the  arch,  a  very  small  change  in 
the  lever  arm  of  F\  or  vertical  distance  between  c  and  ei, 
wouhl  make  a  lar^re  chancre  in  the  value  of  F' . 

These  remarks  are  applicable,  though  in  different  de- 
grees, to  tlie  curve  of  pressure  corresponding  to  all  values 
of  K. 

CALCULATION  OF  THE  MAXIMUM  THRUST  OF  SEMICIRCULAR 
ARCHES  SURCHARGED  HORIZONTALLY  ;  THE  CURVE  OF 
PRESSURE  PASSING  THROUGH  THE  MIDDLE  OF  THE  KEY 
AND  OF  THE  JOINT  OF  GREATEST  THRUST  ;  THE  JOINTS 
OF    EQUAL    LENGTH   THROUGHOUT. 

IIG.  it=the  radius  of  the  extrados;  r=tlie  radius  of 

the  intrados;  ^=the  thickness  of  the  arch  at  the  key; 

Jt  d 

/r=z  — =  1-| — ;  t'  =  the  angle  between  any  johit  and  a 

vertical. 

By  a  course  of  investigation  similar  to  that  explained 
in  the  note  appended  to  equation  (24),  art.  48,  we  find, 
as  the  general  expression  of  the  horizontal  force,  F\  which, 
applied  at  <?,  the  middle  of  the  key,  shall  hohl  in  equili- 
brium "any  segment  a  m  ii  r  ha  on  (?,,  the  middle  of  the 
lower  joint. 

In  like  manner  we  find,  under  the  same  supposition  as 
to  the  curve  of  pressure,  as  a  general  expression  of  ^',  the 
addition  to  tlie  thrust  caused  by  a  surcharge  of  constant 
deptli,  f,  above  the  extrados  of  the  key, 

,r=,.(^+^^y  (61) 

Tlie  maximum  value  of  A'  corresponds  always  to  ^=0 ; 
it  then  becomes, 

.^_ow_^^.  (CA)m 


ACTUAL    THRUST. 


;39 


The  total  thrust  obtained 
in  any  given  case  by  adding 
this  value  of  A  to  tlie  maxi- 
mum value   of  i^',   in    (GO), 
would    involve    an    error    in 
excess,  very  proper   in  light 
arches,  but  not  required  per- 
haps   in    heavy   ones.      The 
words  light  and  heavy,  here 
used,  refer  to  the  proportions 
of  the  arch,  not  to  its  abso- 
lute dimensions. 
'       We  give  below  the  numer- 
ical forms  of  (60)  and  (61) 
corresponding  to  values  of  v  beginning  with  zero  and  in- 
creasing by  5°  to  15%  and  to  v—90°.     By  three  or  four 
substitutions  the  reader  can  oljtain  the  maximum  sum  of 
F'  and  A'  when  r,  /t",  and  t  are  known.     In  practice, 
the  thickness  of  the  aixh  generally  increases  from  the 
key  to  the  reins  or  to  the  springing  line.     In  this  case, 

,7 

/f  =1+--.     The  results  will  be  a  little  m  excess. 
r 

The  sum  of  F'  and  A'  thus  obtained  will  correspond 
to  the  angle  of  maximum  thrust  within  2^  degrees ;  and 
this  is  quite  near  enough. 


.Fig.  32. 


-•M^f^"-') 


v=o° :  F-=r 


1.0019  X  A' -^  +  .33143  X  ^'+1 


i 


—  0.9993: 


A^+1 

Coefficient  of  K%  log.     0.000824. 
"  »  K\    "       T.520395. 


/  1.9962  X^  \ 


340  THEORY    OF    THE    ARCH. 

i.=10°;  F'=rl ^^^ 0.99V4GJ, 


Coefficient  of  K\  log.     0.003325. 
u  u  j^3^    u       T.512918. 


/  1.98481  X  A' \ 


A'+l  / 

Coefficient  of  K\  log.     0.007088. 
„  u  X*,    "      T.500361. 


/1.0G593x7Jr\ 
^^20°;  F'—r'i 


1.02834  X  ^'+0.303785  x  K-^r\     ,  „,^,,\ 

Coefficient  of  K\  log.     0.012135. 
"  "  A"^   "       T.4825G7. 


/ 1.93909  X  A' \ 


Coefficient  of  K\  log.     0.018058. 
"  "  A'\    "       T.459319. 


A'  =  rt{ 


/1.90G31  xA'\ 

Coefficient  of  A'',  log.     0.024490. 
u  a  x\    "       T.430298. 

/1.86G03xA'\ 

/  1.07408  x  A''  +  0.2483G  X  A'='  +  | 
1-1=35°;  F'  =  rH 


A^+1 

Coefficient  of  K\  log.     0.03 1 035. 
«  "   A'^    "       T.395084. 


—  0.9C871 


/1.81915xA'\ 


NUMERICAL    FORMS.  341 


.=40°;  ^■^,.^Lg8j8>'^"+°|lS«x^-±l_0.85905); 


Coefficient  of  K\  log.     0.03'7273. 
"  "   /f',     "       T.353105. 


A'=ir( 


/ 1.76604  xA"\ 


,_      ^,       ,/ 1.10355  xA'»  +  0.20118x^'+|  \ 

v  =  45°  ;  i^  =r^l ^^^-- Z^_0.94806  j. 


Coefficient  of /r-,  log.     0.042792. 
"  A^^    "       T.303594. 


A'=rt 


/1.V0711  x/ir\ 

„      ^,       ,/ 1.1148  x/r-  +  0.1'7599x^'+^  ^      ' 

v  =  50°;  A"  =  r^( ^-— ^^^-0.935715 


Coefficient  of  K^,  log.     0.04Y199. 
"  "  /f^    "       T.245498. 


/ 1.64279  xA^\ 
A=^rt[ ). 


„„      ^.       „/l.l223x  ^'  +  0.15043  x^'+^  \ 

t'  =  55°  ;  F  =r/ kJ?\ -^-0.922004  J  ; 

Coefficient  of  A"',  log.     0.050106. 
"  K\    "       T.177328. 
/1.57358X  A'X 

v  =  60°  ;  A^  =?•-!  -^ — Tf       ^^  —  0.90690  J  ; 

.^H^l^!+?;1101^^!±i_0.89037)  ; 

Coefficient  of  K\  log.     0.049995. 
"  "   K\    "       T.000885. 


v  =  65°;  A'3=?' 


/  1.42262  X  A"  \ 


342  THEORY     OF    THE    ARCH. 

„=:0«;  r=r(Hi'^=i^^±|5!?i2i£!±i-0.872404) 

Coefficient  of  X*,  log.     0.046310. 
u  «  ^"3^    u       '2.883061. 

/  1.34202  X  A' \ 

Coefficient  of  X\  log.     0.039778. 
"  /l^    "       "2.734794. 
/  1.25882  X  A' \ 

/  A'^  4-  '^  \ 

r=90°;  A" ^r'/ -^-^-0.78540  J  ; 


From  (GO)  we  have  calculated  tahle  DD,  giving  what 
we  may  regard  as  tlie  actual  tlirnst  of  the  semicircular 
arch,  surcharged  horizontally,  uuder  the  conditions  ex- 
pressed at  tlie  head  of  this  article.  Tlie  tahle  also  gives 
the  maximum  effect  of  the  surcharge  of  any  constant 
dei»th,  f,  aT)ove  the  suuHiiit  of  tlie  arch.  In  drawing  up 
the  table  we  have  reduced  (GO)  to  its  numerical  form  for 
every  value  of  v,  in  whole  numl)ers,  from  40°  to  75°,  inclu- 
sive. But  knowing  the  resulting  maximum  thrusts  to  be 
somewhat  greater  than  they  need  l)e  in  heavy  arches,  we 
have  su})posed  t'=::60°  for  all  values  of  A'^ exceeding  1.22. 

Being  once  on  the  track  we  have  generally  found  the 
maximum  value  of  i^',  corresponding  to  any  particular 
value  of  /if,  by  three  or  four  substitutions. 


NUMERICAL    FORMS.  343 


Column  1  ejives  the  value  of  K:=l-\ — • 


2?' 
"         -y,  or  ratio   of  tlie  span  to 

tlie  thickness, 
angle   of    maximum   tlirust   down  to 

/ir=1.22,  'y=r45°;  below  that,  v  is 

assumed  at  60°. 
maximum  value  of  F'  down  to  K=^ 

1.22  ;  below  that  the  value  of  F' 

corresponding  to  -^=60°. 
for  the  purpose    of   comparison,  from 

table  D,  the  maximum  and  actual 

thrust  in  the  case  of  rupture  and 

fall. 

Tjl 

S=^^ri  01*  ratio  of  these  two  thrusts, 

properly  the  coefficient  of  stability. 

the  value  of  A^  or  maximum  effect  of 
the  surcharge,  v^^O. 

for  the  purpose  of  comparison,  from 
table  F^  the  values  of  A^^  or  max- 
imum effect  of  the  surcharge  in 
case  of  rupture  and  fall. 

6'=-7— ,  or  ratio  of  these  two  effects. 
A2 

11 


344 


THEORY     OF    THE    ARCH. 


CALCULATION  OF  THE  MAXIMUM  THRUST  OF  THE  ROOF- 
SHAPED  SEMICIRCULAR  ARCH  ;  THE  CURVE  OF  PRP:SSURE 
PASSING  AT  J  THE  LENGTH  OF  THE  JOINT  FROM  THE 
EXTRADOS  AT  THE  KET,  AND  FRO^I  THE  INTRADOS  AT 
THE   JOINT   OF    GREATEST   THRUST. 


Fig.  88. 


117.  i?=the  radius  of  the  extrados ;  7'=tlie  radius  of 
the  intrados ;  (/=:the  thickness  of  the  arch  at  the  key ; 

/r= —  =  1H —  ;  /=the  anf>;le  between  the  roof  and  a  ver- 
;•  r  ° 

tioal ;  f =the  angle  between  any  joint  and  a  vertical. 

By  a  course  of  mvestigation  similar  to  that  referred  to, 

ai-t.  116,  we  find,  as  the  general  expression  of  the  hoi'izontal 

force,  F'  which,  applied  at  c^  on  the  central  joint,  a  ^,  c  b= 

•^«^,  shall  hold  in  equilil)rium  any  segment,  a  7nnp^a^ 

on  ^1,  a  point  of  the  lower  joint  ??i ;?,  9JI  Ci=^^  m  ??, — 


( 


K'  X  2(2-sin.(/4-?'))-7r''(l  -sin.(/+i>))     ?'(A'+2),       1 


sin.  / 


+ 


sin.  V        cos.^^tA    (62) 


■)■ 


K(2  —  COS.  ?')-{- 1  —  2  COS.  V 

In  like  manner  we  find,  under  tlie  same  supposition  as 
to  the  curve  of  pressure,  as  a  general  expression  of  A\  the 
addition  to  the  thrust  caused  by  a  surcharge  of  the  con- 
stant depth  t,  above  the  tangent  roof,  D  7?, — 


THE    MAGAZINE     ARCH.  345 

A'=h'tKmi.'  vy^ ^"^^^—^ (^^) 

ii  (2  —  COS.  V)  +  1  —  2  c< )s.  -y 

of  wliicli  the  maximum  value  is 


corresponding  to  an  angle  whose  cosine  is 

2ir+l-|/3A"2-3  ...X 

cos.  v= -~X4^ ^  ^ 

From  (62),  (64),  and  (65),  we  have  calculated  taLle 
FF,  giving,  directly  or  by  proportional  parts,  under  the 
conditions  expressed  at  the  head  of  this  article,  the  actual 
thrust  in  all  the  isolated  magazine  arches  in  common  use. 
In  drawing  up  the  table  we  have  reduced  (62)  to  its 
numerical  form  for  values  of  /  res2:)ectively  equal  to  60*^, 
55°,  50°,  and  45°,  and  for  values  of  v  increasing  by  2i°, 
from  30°  to  60°,  that  is,  as  far  as  necessary,  both  ways,  to 
ascertain  the  maximum  thrust. 

The  results  under  each  value  of  7,  are  not  exactly  the 
maximum  thrust,  but,  in  general,  a  little  less ;  the  differ- 
ence, however,  is  practically  nothing. 

EXPLANATION   OF   TABLE   FF. 

Column  1  fi-ives  the  value  of  Ar=l+-. 

"  1,  under  each  value  of  /,  gives  the  angle  of  great- 
est thrust. 

"  2,  under  each  value  of  /,  gives  the  decimal  C; 
i^=the  thrust=i;-'(7. 

"  3,  under  each  value  of  I,  gives  the  coefficient  of 
staljility,  ^,  or  ratio  of  the  actual  thrust  to  the 
diminished  thrust  at  the  moment  of  rupture 
and  fall,  the  latter  being  oV)tained  from  tal)le  F. 

"  1,  under  "  surcharge,"  gives  the  angle  of  maximum 
thrust  of  the  surcharge. 

"  2,  under  "  surcharge,"  gives  the  decimal  C;  A  = 
rtC=the  maximum  effect  of  the  surcharge. 


34G  THEORY     OF    THE     ARCH. 

EEMARKS. 

It  will  be  seen  that  the  angle  which  rendei's  the  effect 
of  surcharije  a  maximum,  diffei's  Liit  a  few  de<?rees  from 
the  angle  of  maximum  thrust  in  the  arch  proper ;  conse- 
({uently  the  error,  in  excess,  which  we  commit,  by  adding 
the  two  maxima  together,  and  taking  their  sum  as  the 
actual  thrust  of  the  arch  and  its  load,  is  exceedingly  small, 
and  the  table  may  be  regarded  as  practically  exact. 

It  will  also  be  seen  that  the  coefficient  of  stability,  5,  is 
nearly  the  same,  for  the  same  values  of  /r,  in  all  the 
arches ;  consequently,  we  can  obtain,  from  table  F,  the 
actual  thrust,  on  the  conditions  announced  at  the  head  of 
this  article,  for  values  of  I  some  degrees  above  60°,  by 
multii)h'ing  the  thrust,  computed  from  that  table,  by  the 
value  of  h  found  opposite  the  given  value  of  K  in  table 
FF. 

Tlie  maximum  effect  of  surcharge,  given  l)y  the  last 
column  of  FF,  is  independent  of  /,  and  the  same  in  all 
arches. 

For  rules  for  using  table  FF,  see  rules  for  using  table 
F,  arts.  01,  6-2. 

CALCULATION  OF  THE  THRUST  OF  THE  SEMI-CIRCULAR  ARCH 
SURCHARGED  HORIZONTALLY  ;  THE  CURVE  OF  PRESSURE 
PASSING  AT  J  THE  LENGTH  OF  THE  JOINT  FROM  THE 
EXTRADOS  AT  THE  KEY,  AND  FROM  THE  INTRADOS  AT 
THE   JOINT    OF   GREATEST   THRUST. 

118.  This  is  a  particular  case  of  the  roof-shaj^ed  arch 
discussed  in  art.  117.  We  need  not  repeat  that  dis- 
cussion. Formulae  (62),  (63),  (64),  and  (65),  remain  the 
same.  In  the  first  it  is  only  necessary  to  make  /=90° 
Avliieh  reduces  (62)  to 

„,_.   ,   .    ,    J '  s'"-^        ^Q'^-H^    (62)' 

^  -*'■  "'"■  ^'  A'(2-cos.^')-i-l-2co.s.l; 


TABLE     DDD.  347 

From  this  formula  and  from  other  sources,  we  have  cal- 
culated table  DDD,  giving  the  maximum  or  actual  thrust 
of  the  arch  in  question  under  the  conditions  stated  above. 

EXPLANATION    OF   TABLE   DDD. 

The  1st  column  gives  the  value  of  ^=1-] — -j   (/=the 

thickness  at  the  key. 

The  second  column  gives  the  decimal  C ;  F=zrC^i\\Q 
thrust  on  the  condition  stated  at  the  head  of  the  table. 

The  3d  column  gives  the  decimal  C;  ^^zr^C— the 
addition  to  the  thrust  caused  by  a  surcharge  of  the  con- 
stant depth  t  above  the  key. 

F-\-A=^i\\Q  entire  thrust,  with  an  excess  arising  from 
adding  two  maxima  together. 

The  4th  column  gives  the  joint  of  rupture  or  angle 
of  greatest  thrust  on  the  supposition  of  actual  rupture  and 
fall,  from  table  D. 

The  5th  column  gives  the  angle  of  greatest  thrust  on 
the  supposition  that  the  curve  of  pressure  is  at  J  the 
length  of  the  joint  from  the  extrados  at  the  key,  and  from 
the  intrados  at  the  joint  of  greatest  thrust,  calculated  at 
intervals  oi^\  degrees. 

The  6th  column  gives  the  angle  of  greatest  thrust  on 
the  supposition  that  the  curve  of  pressure  is  at  the  center 
of  the  joints  of  the  key  and  of  greatest  thrust,  taken 
from  table  DD.     Below  ^"=1.22  the  angle  is  not  given. 

The  7th  column  gives  the  coefficient  of  stability,  (5,  or 
the  ratio  of  the  actual  thrust,  on  the  conditions  stated  at 
the  head  of  the  table,  column  2,  to  the  calculated  thrust  at 
the  instant  of  rupture  and  fall,  table  D. 

The  8th  column  gives  the  coefficient  of  stability  on  the 
supposition  that  the  curve  of  pressure  passes  through  the 
micldle  of  the  joints,  from  table  DD. 


34S  TUEOKY  OF  THE  ARCH. 


REMAKKS  OX  TABLE  DDD. 


Coliiiiin.-!  G  aucl  8  liave  been  taken  from  tal)le  DD. 
Tliey  are  added  here  that  the  reader  may  see  at  a  glance, 
how  the  anQ:les  of  s^reatest  thrust  and  the  thrusts  them- 
selves  vary  Avitli  the  suppositions  Avhieh  we  make  upon  the 
curve  of  pressure. 

The  greatest  value  of  b  in  column  V,  is  1.92;  in  column 
8,  it  is  2.5i). 


CALCULATION  OF  '  THE  THRUST  OF  SEGJilENTAL  ARCHES, 
SURCHARGED  HORIZONTALLY,  THE  CURVE  OF  PRESSURE 
PASSING  THROUGH  THE  IVIIDDLE  OF  THE  KEY,  AND  THE 
]MIDDLE  OF  THE  JOINT  OF  GREATEST  THRUST,  AVHICH  IS 
GENERALLY    AT   THE   SPRINGING    LINE. 

119.  If  the  semi-angle  at  the  center  be  as  great,  or 
nearly  as  great,  as  the  angle  of  maximum  thrust  in  table 
DD,  art.  116,  we  shall  find  in  that  table  the  actual  thrust 
on  the  condition  announced  above.  This  table  may, 
indeed,  be  used  in  all  cases  without  any  great  error;  for  we 
have  shown  that  the  curve  of  pressure,  passing  through  the 
middle  of  the  joint  at  the  key  and  the  middle  of  the  joint 
of  greatest  thrust,  will  continue  near  the  centers  of  all 
intermediate  joints.  The  error  will,  of  course,  be  always 
in  excess. 

The  exact  thrust,  when  it  is  less  than  that  given  in 
DD,  will  be  obtained  from  equation  (GO)  art.  IIC,  when 
we  have  substituted  for  the  constants  which  enter  into 
that  fornnila,  their  known  values. 

Tlie  effect  of  a  surcharge  of  constant  dej)th  may  also 
be  obtained  from  table  DD,  with  an  error,  always  in 
excess. 


SEGMENTAL    ARCHES.— TABLE     EE.  349 


CALCULATION  OF  THE  THRUST  OF  SEGJIENTAL  ARCHES,  SUR- 
CHARGED HORIZONTALLY,  THE  CURVE  OF  PRESSURE 
PASSING  AT  J  THE  LENGTH  OF  THE  JOINT,  FROM  THE 
EXTRADOS  AT  THE  KEY,  AND  FPvOM  THE  INTRADOS  AT 
THE  JOINT  OF  GREATEST  THRUST,  WHICH  IS  GENERALLY 
AT   THE    SPRINGING    LINE. 

120.  x^otatiou:  5=the  span;  /=:tlie  rise;  r=t\\e. 
radius  of  tlie  intrados;  d—the  thickness  of  the  arcli  at 

thekey;  K=l-\ — ;   6^=  the  decimal  in  the  first  column 

under  each  value   of  v;    i^'^the  tlirust=7'^(7;    t'=the 
semi-anojle  of  the  whole  arch. 

If  one  half  the  angle  subtended  by  the  given  arch  be 
as  great,  or  nearly  as  great,  as  the  angle  of  maximum 
thrust  in  column  5,  table  DDD,  we  can  obtain  the  required 
thrust  directly  from  that  table.  But  if  this  half-angle,  v, 
be  less  than  the  angle  of  maximum  thrust,  substitute  its 
known  value,  and  the  value  of  X in  (62)':  the  resulting 
value  of  ^'  will  be  the  actual  thrust  on  the  condition 
stated  above. 

To  obtain  the  effect  of  a  surcharge  of  the  constant 
depth  t  above  the  key :  if  ^  be  as  great,  or  nearly  as  great, 
as  the  angle  which  renders  the  effect  of  surcharge  a  maxi- 
mum, see  last  column  but  one  of  table  FF,  the  required 
addition  will  be  obtained  from  the  last  column  of  that 
table,  or  from  the  4th  column  of  table  EE. 

But  if  V  be  less  than  the  angle  in  question,  the  required 
effect  of  surcharge  must  be  computed  from  formula  (63). 
The  sum,  -F+^,  as  usual,  will  be  the  entire  thrust. 

Table  EE  gives,  either  directly  or  by  proportional 
parts,  the  actual  thrust  of  segmental  arches  in  common 
use. 


350 


THEORY    OF    THE    A  RGB. 


EXPLANATION   OF  TABLE   EE. 

Column  1  2:1  ves  tlie  value  of -2  =  14 — . 

"  1,  under  each  value  of  v^  gives  the  decimal  C ; 
i^=:r2^=the  thrust  of  the  arch  loaded  up 
to  the  level  of  the  extrados  at  the  key. 

"  2,  under  each  value  of  v^  gives  the  ratio,  ^,  of  this 
actual  thrust  to  the  thrust  of  the  same  arch 
at  the  moment  of  ruj)ture  and  fall,  taken 
from  table  E'. 

"  3,  under  each  value  of  7;,  gives  the  decimal  C; 
A^rtC—\\\Q  addition  to  the  thrust  caused 
by  a  surcharge  of  the  uniform  depth  t. 

ELLIPTICAL  ARCHES. 

121.  In  the  first  part  of  this  paper,  section  V.,  art.  90, 
and  following,  we  have  shown  how  to  obtain  the  ultimate 
thrust  of  the  elliptical  arch,  loaded  and  unloaded.  The 
actual  thrust  may  be  found  in  a  similar  manner. 


ELLIPTICAL  ARCHES  SURCHARGED  HORIZONTALLY. 

Let  us  compare  the 
given  arch  witli  a  cir- 
cular arch,  surcharged 
in  like  manner  hori- 
zontally, having  the 
same  span,  and  a  thick- 
ness at  t lie  key  as  much 
greater  than  the  thick- 
ness of  the  elliptical 
arch  as  the  half-span  is 
greater  than  the  rise. 
Through  <?,  the  middle 
of  a  h^  and  Oa,  the  mid- 


Fic.  at 


ELLIPTICAL    ARCHES.  351 

die  of  A  B^  draw  tlie  ellipse  c  c^  c^  similar  to  the  iiitra- 
dos.  This  curve  will  cut  the  several  joints  of  the  elliptical 
arch  near  their  central  points,  and  will  never  pass  witliin 
the  inferior  limit,  one-third  the  length  of  a  joint  from  the 
intrados,  unless  Ave  give  an  unnecessary  extension  of  these 
joints  at  the  reins.  Draw  the  curve  g  c\  <?3,  dividing  into 
equal  parts  the  joints  of  the  auxiliary  circular  arch,  sup- 
posed here  to  be  of  equal  thickness  throughout.  Drawing 
any  vertical  lJ  tp^  let  /S'rzrthe  surface  m  t  h  »,  /iS"=the  surface 
m'  t'  h'  a\y=:^GX^  or  vertical  distance  between  c  and  (^21 
y' =zc' w\  p=^t}iQ  horizontal  distance  between  the  vertical 
t  p  and  the  center  of  gravity  of  S^  j9'=the  corresponding 
distance  in  the  circular  arch. 

The  horizontal  force  i^  which,  applied  at  <?,  shall  hold 
in  equilibrium  the  surface  S^  in  relation  to  ^25  ^s  'the  center 

of  rotation,  is  i^= —  :   in  like  manner,  we  have  ^'= 

y 
S'xp' 

But  wherever  the  vertical  be  drawn,  the  depths  m  % 
m'  t\  stand  in  the  constant  relation  of  the  rise  to  the  half- 
span,  or  /  to  r  ;  the  surfaces  /S,  8\  and  the  lever-arms  y 
and  y'  stand  in  the  same  relation ;  the  centers  of  gravity 
of  8  and  S'  are  on  the  same  vertical  line,  so  that  p)=^p'- 
"We  have,  therefore, 

8\8''.'.y'.y\  and  ^^§^^  or  F:=F'. 

These  surfaces  having  the  same  thrusts  wherever  the 
vertical  be  drawn,  their  maximum  thrusts  will  also  be  the 
same  ;  and  we  arrive  at  this  result : 

The  actual  thrust  of  an  elUptkal  arch  sustaining  a 
load  of  masonry  or  of  equal  tueight  with  masonry^  rising 
to  the  horizontal  linejangent  to  the  extrados  at  the  Jcey^  is 
nearly  equal  to  the  thrust  of  the  semiciraular  arch^  loaded 
in  nice  manner^  having  the  same  span^  and  a  thickness  at 


352  THEORY     OF    THE     ARCH. 

tlie  hey  as  much  greater  tlian  tlie  thicl-nes-s  of  tlie  elliptical 
arch^  as  the  JiaJf-q)an  is  greater  than  the  rise. 

We  shall,  tlierefore,  be  able  to  obtain  immediately 
from  taT)le  DD,  the  actual  thrust  of  the  elliptical  arch. 

Strictly  speaking,  a  slight  correction — addition — would 
be  necessary,  as  we  have  disregarded  the  influence  of  the 
small  surfaces  c^  n  r  t^  c'g  n  r  t'  /  but  on  the  other  hand 
we  have  provided  for  some  exaggeration  of  the  thrust,  l)y 
supposing  the  curve  of  pressure  to  pass  through  the 
middle  of  the  key,  and  near  the  middle  of  the  joint  of 
greatest  thrust. 

Example.  Central  arch  of  the  London  Bridge.  /•= 
half-span =Y 6';  /=the  rise=38';  rZ=tlie  thickness  at  the 
key =5' ;  Z^=the  thickness  of  the  auxiliary  circular  arch  = 

r  D 

-Xd=l0';  /r=l+— =  1.1316;  hence  from  the  4th  col- 
umn of  table  DD,  by  proportional  parts,  i'^y'X. 16723  = 
966.45  cubic  feet.  Suppose  a  surcharge  one  foot  deep 
above  the  key,  ^=1';  the  corresponding  surcharge  of  the 

auxiliary  circular  arch  is  -^"=2',  and  its  effect  is,  Ttli  col- 
umn of  table  DD,  .l  =  2'x76'xl.0617  =  161.38. 


THE  COEFFICIENT  OF  STABILITY. 

122.  In  the  first  part  of  this  paper,  we  have  shown 
how  to  find,  by  tables  or  calcuhition,  the  actual  thrust  of 
most  arches  at  the  supposed  moment  of  rupture  and  fall, 
when  all  the  forces  in  the  system  act  upon  three  points  or 
edges  of  masonry  ;  viz.,  the  extrados  at  the  key,  the  intra- 
dos  at  the  reins,  and  the  exterior  lower  edge  of  the  semi- 
arch  or  pier.  We  have  shown,  art.  109,  that  this  ultimate 
thrust  Is  also  the  minimum  or  least  possible  thrust  that 
can  ever  exist  in  the  arch. 


COEFFICIENT    OF    STABILITY.  353 

This  minimum,  Audoy,  Poncelet,  and  others,  multiply 
by  what  tliey  call  the  coefficient  of  stability :  2,  or  some 
smaller  number,  according  to  the  proportions  of  the  arch ; 
• — and  they  determine  the  thickness  of  pier  on  the  condi- 
tion that  the  resultant  of  the  thrust  thus  increased,  and  of 
the  weight  of  the  semi-arch  and  pier,  shall  pass  through 
the  exterior  edge  of  the  base  of  the  pier. 

The  value  of  this  coefficient  was  determined  by  an 
examination  of  a  powder-magazine  of  Vauban,  which,  hav- 
ing stood  the  test  of  ages,  w^as  presumed  to  have  all  the 
necessary  elements  of  stability.  The  value  of  the  co-effi- 
cient thus  determined,  was  found  to  be  about  2  ;  and  this 
is  evidently  applicable  to  all  similar  structures,  that  is,  to 
structures  identical,  or  nearly  so,  with  the  magazine  of 
Vauban,  in  all  their  proportions,  and  only  different  in 
their  absolute  dimensions.  The  rule  has  been  of  great  ser- 
vice, for  it  so  happens  that  most  magazines  have  been 
modeled  after  that  of  Vauban. 

But  the  idea  was  entirely  empirical,  and  was  so  under- 
stood by  its  authors.  The  co-efficient  of  stability  thus 
defined  and  used,  is  by  no  means  a  correct  index  or  mea- 
sure of  the  stability  of  any  pier.  In  piers  of  great  height, 
it  will  generally  give  results  too  small, — in  piers  of  small 
height,  results  too  large  ;  nor  is  it  sure  to  give  correct 
results  in  any  work  not  strictly  similar  to  that  of  Vauban 
from  which  the  rule  was  deduced.  The  proper  value  of 
this  coefficient  will  change,  not  only  with  every  variation 
of  the  proportions  of  the  arch,  but,  in  the  same  arch,  with 
every  increase  or  reduction  of  the  height  of  the  pier.  Nor 
is  it  possible  to  know  this  proper  value  without  first  learn- 
ing the  actual  thrust  of  the  well-established  arch;  and 
when  we  have  attained  this  knowledge,  we  have  only  to 
make  proper  use  of  it :  the  coefficient  has  become  useless. 
We  know  the  thrust  itself 

As  it  is  evident  that  the  curve  of  pressure  in  the  arch 
should  be  everywhere  traced  between  two  other  curves 


354  THEORY    OF    THE    ARCH. 

which  divide  the  joints  into  three  ecjual  parts, — so,  in  the 
pier,  tliis  curve  must  lie  between  corresponding  lines ;  and 
at  the  base,  where  it  approaches  nearest  to  the  exterior 
face,  it  must  not  come  nearer  than  J  the  length  of  the 
lower  joint,  or  J  the  thickness  of  pier.  This  conclusion  is 
inevitable,  if  we  admit  the  principle  of  Navier  as  to  the  dis- 
tribution of  the  pressures  upon  the  joints ;  and  this  })rin- 
ciple  is  generally  admitted,  we  believe,  by  engineers. 

123.  Let  us  examine  the  magazine  of  Yauljan  anew, 

fis:.  13.     Its  dimensions  are  :  Radius  of  the  intrados  =  r= 

R 
12'.r>0  ;  radius  of  the  extrados=i?=15'.50  ;  /r=-^=1.24  ; 

inclination  of  the  roof  to  a  vertical =7=49°  7'  17"  ;  height 
of  the  pier  from  the  base  to  the  springing  line=/?  =  8'; 
thickness  of  pier  given  by  Vauban=:8';  on  the  exterior 
face,  counterforts  6'  long  and  4'  deep,  with  intervening 
spaces  of  12' ;  whole  thickness  of  pier  and  counterfort =12'. 

The  ultimate  thrust  of  this  arch,  arts.  48,  63,  is  F— 
r'^X  0.2294.  But  we  learn  from  table  FF  that  its  least 
actual  thrust,  consistent  with  the  condition  that  no  joint 
shall  open  at  either  the  extrados  or  intrados,  is  ^•'^X  0.3496. 
Substituting  this  value  in  formula  (31-^)'  art.  66,  we  obtain, 
as  the  least  thickness  of  a  solid  pier  whose  lowest  joint 
T\ill  not  open  on  the  inside,  10'.56.  The  mean  thickness 
of  Vauban's  wall  is  9'.33,  and  the  thickness  given  by  the 
rule  of  Audoy,  fc2,  formula  (31)',  art  66,  is  9'.23. 

We  come  at  once  to  this  unexpected  result, — that  the 
rule  of  Audoy  does  not  in  this  case  give  a  pier  of  sufficient 
thickness  to  withstand  the  thrust  of  the  arch  without  any 
openings  of  the  joints.  We  are  authorized  and  comi)elled 
to  conclude,  that  the  magazine  of  Vauban  derived  some 
additional  stability  from  adhesion  of  mortar  in  the  arch 
or  pier,  or  both. 

During  the  construction  of  an  arch  it  is  almost  impos- 
sible  to   prevent   cracks   or   openings  at  the  reins,  and 


MAGAZINE    OF    VAUBAN.  355 

althoiigli  they  may  close  on  the  removal  of  the  center,  still 
the  adhesion  of  mortar  must  be  forever  impaired  or  de- 
stroyed. It  would  be  unwise  to  rely  upon  any  adhesion  of 
mortar  in  the  arch. 

On  the  other  hand,  the  pier  is  subject  to  no  disturbance 
during  its  construction  ;  its  mortar  has  time  to  set ;  and 
w^e  may  in  some  cases  rely  with  confidence  upon  this  ele- 
ment of  stability. 

The  calculation  of  the  effective  resistance  of  the  piers 
in  Vauban's  magazine  is  complicated  by  the  existence  of 
abutments.  Let  us  still  assume  that  the  lowest  joint, 
under  the  action  of  the  horizontal  thrust,  is  on  the  point 
of  opening  at  its  inner  edge,  where  the  pressure  is  conse- 
quently zero,  and  that  the  pressure  increases  unifoi'mly 
from  that  point  to  the  exterior  edge  of  the  abutments. 
The  mean  j)ressure  will 
be  represented  by  the 
ordinates  of  the  oppo- 
site Figure,  and  the 
point  of  application  of 
the  resultant  of  all  the  pressures  will  be  at  C,  B  0=5'. 2 55. 

The  mean  center  of  gravity  of  the  pier  is  Y'.143  from 
JB,  the  exterior  edge.  Taking  the  moment  of  the  pier  and 
of  the  semi-arch  in  relation  to  0^  we  find  the  thrust,  acting 
horizontally  one  foot  below  the  extrados  at  the  key,  which 
stands  in  equilibrium  with  these  elements  of  resistance,  to 
be  52.70  =  /-^X  0.3373,  less  than  the  actual  thrust  before 
given  by  7-^x0.0123.  The  supposition  of  a  very  slight 
adhesion  of  mortar  upon  the  base  of  the  pier,  viz.,  1^ 
pounds  per  square  inch,  not  more  than  a  fifteenth  part  of 
the  adhesion  of  good  mortar,  will  equalize  the  two  thrusts. 

It  thus  appears  that  the  magazine  of  Vauban  illustrates 
and  confirms  in  a  remarkable  manner  the  new  theory  of 
the  arch.  Its  piers  present  almost  precisely  the  resistance 
which  this  theory  requires. 


356  THEORY     OF    THE    ARCH. 

COEFFICIENT    OF   STABILITY NEW    DEFINITION. 

124.  The  coefficient  of  stability  is  the  ratio  of  tlie 
actual  thrust  of  tlie  well-established  arcli  to  the  ultimate 
thrust  existing  at  the  moment  of  rupture  and  fall,  or  the 
quotient  of  the  former  divided  by  the  latter. 

This  ultimate  thrust  for  most  arches  likely  to  occur  in 
practice,  is  given  l)y  the  tables  which  belong  to  the  first 
part  of  this  paper,  viz.,  A,  C,  D,  E,  E',  F,  and  G. 

If  we  had  tables  equally  extensive  of  the  actual  thrust, 
we  should  no  longer  need  this  coefficient.  But  at  present 
such  is  not  the  case.  M.  Cavallo  has  given  an  extensive 
table  of  the  actual  thrusts  of  semicircular  arches,  based 
on  the  supposition  of  vertical  joints,  and  of  a  sui-charge 
l:)Ounded  horizontally  at  top  with  a  depth  on  all  vertical 
lines  bearing  a  constant  ratio  to  the  depth  of  the  arch 
proper  on  the  same  lines.  And  we  have  given  in  this  pa- 
per tables  of  the  actual  thrusts  of  the  unloaded  circular 
ring,  AA,  art.  114  ;  tlie  magazine  or  roof-shaped  arch,  FF, 
art.  117  ;  the  semicircular  arch  surcharo:ed  liorizontally, 
DDD,  ai-t.  118,  the  segmental  arch  surcharged  horizontally, 
EE,  art.  120 ;  aU  based  on  the  supposition  of  joints  of 
equal  length  perpendicular  to  the  intrados,  and  of  a  curve 
of  pressure  at  the  key  ^  the  length  of  the  joint  from  the 
extrados,  and  at  the  j(jint  of  greatest  thrust  -i-  the  length 
of  the  joint  from  the  intrados  ;  also,  a  table,  DD,  art.  116, 
of  the  actual  thrusts  of  semicircular  arches  surcharged 
horizontally,  with  joints  as  above,  and  with  a  curve  of 
pressure  passing  thruugh  the  middle  of  the  key-stone  and 
weakest  joint. 

We  have  given  in  all  of  these  tables  the  value,  ^,  of 
the  coefficient,  iis  defined  above,  to  show  at  a  glance  liow 
the  actual  compares  with  the  ultimate  tlirust  wlien  both 
are  known,  and  to  exhibit  a  guide  which  may  lead  to  the 
former  when  the  latter  only  is  known. 

For  instance,  we  learn,  from  table  FF,  that  the  value 


COEFFICIENT    OF    STABILITY.  857 

of  5,  correspondmg  to/r=1.20,  is  1.47  when  7=45°;  1.46 
when  7=50°;  1.46  when  7=55°;  1.45  when  7=60°;  and 
we  learn  from  tahle  DDD  tliat  the  vahie  of  ^  is  1.47  wlien 
7:"=  1.20  and  7=90°;  we  may  thence  conclude,  by  anal- 
ogy, that  8  does  not  exceed  1.47  for  roofs  of  any  inclina- 
tion between  a  horizontal  and  45°.  In  like  manner  we 
learn  the  value  of  h  for  other  values  of  If,  and  may  deduce 
the  actual  thrust  from  table  F,  when  the  given  value  of  7 
does  not  place  the  case  in  table  FF. 

In  all  this  we  use  the  ultimate  thrust  as  a  convenient 
and  almost  indispensable  standard  of  comparison ;  not 
only  because  that  thrust,  as  already  remarked,  has  been 
extensively  calculated  and  published,  but  also  because  it 
can  be  determined  with  great  accuracy  by  experiments 
upon  models.  The  actual  thrust  cannot,  with  equal  accu- 
racy be  thus  determined. 

Kegarding  this  ultimate  thrust  as  the  standard,  we  may 
with  propriety  call  the  ratio  by  which  it  must  be  multi- 
plied to  give  the  actual  thrust,  the  coefficient  of  staUlity. 


DISCUSSION  OF  THE  COEFFICIENT  OF  STABILITY,  OR  RATIO 
OF  THE  ACTUAL  TO  THE  ULTIMATE  THRUST,  WHEN  THE 
CURVE  OF  PRESSURE  LIES,  AT  THE  KEY,  J  THE  LENGTH 
OF  THE  JOINT  FROM  THE  EXTRADOS,  AND  AT  THE  REINS 
J   THE    SAME   DISTANCE   FROM   THE   INTRADOS. 

125.  Semicircular  arches  surcJiarged  horizon  tally. — We 
learn  from  column  7,  talkie  DDD,  that  this  ratio,  begin- 
ning with  1  for  7-^=1,  gradually  increases  by  nearly  arith- 
metical differences,  to  its  maximum,  1.92,  corresponding  to 
7r=1.35.  It  then  begins  to  diminish,  and  would  finally 
become  1  again. 

The  magazine  arch. — We  learn  from  table  FF,  that 
the  value  of  ^  constantly  increases  from  7i'=1.15  to  K= 


358  THEORY     OF    TUE    ARCH. 

1.40.    It  attains,  in  fi\ct,  its  greatest  value  when  /i^  slightly 
exceeds  I.-IO. 

The  value  of  5  corresponding  to  the  usual  values  of  K^ 
say  from  7r=1.15  to  7^=1.30  is  nearly  the  same  in  each 
horizontal  colunm.  We  hence  infer  the  constancy  of  5  for 
intermediate  values  of  /,  and  for  values  somewhat  less 
than  45°  or  more  than  60°. 

Comparing  together  tables  FF  and  DDD,  we  see  that 
I  up  to  /r=1.18,  is  less  with  the  horizontal  than  with  the 
inclined  roof;  that  it  is  about  the  same  with  all  roofs  from 
71^=1.18  to  7^=1.24 ;  and  that  above  A"=1.24,  it  predom- 
inates more  and  more  with  the  horizontal  roof  up  to  K— 
1.35.     Above  that,  its  predominance  diminishes. 

Segmental  arches  surcliarged  liorizontaUy. — Table  EE. 
The  coefficient  5,  begi^ining  with  1  for  K=\,  gradually 
increases  under  each  value  of  v^  or  each  ratio  of  the  span 
to  the  rise,  to  its  maximum,  corresponding  to  values  of  K 
varying  from  1.34  to  1.07.  The  maximum  itself  dimin- 
ishes from  1.938  under  ^:=4/,  to  1.788  under  <s=8/,  and 
then  increases  to  1.938  under  s—lQf. 

The  great  variation  of  the  coefficient,  ^,  from  the  low- 
est to  the  higher  values  of  TT,  viz.,  nearly  from  1  to  2, 
proves  the  great  inaccuracy  of  the  rule  of  Audoy,  as  used 
by  him  and  others.  Even  with  the  same  values  of  K^  this 
coefficient  varies  largely:  opposite  7r=1.07,  from  1.177 
to  1.938;  opposite /ir=  1.03,  from  1.077  to  1.506;  oppo- 
site /ir=1.34,  from  1.938  to  1. 

We  have  not  given  the  ratio  of  the  actual  and  ultimate 
effects  of  surcharge  upon  the  thrust ;  ])ut  in  tables  F  and 
FF  have  rnven  the  effects  themselves,  from  w^hich  the  cor- 
responding  ratios  may  be  easily  determined.  In  general 
they  differ  but  little  from  the  coefficient  of  stability  of  the 
arches  to  which  they  belong.  This  last  remark  is  also 
applicable  to  segmental  arches. 


THICKNESS    OF    TIER.  359 


THE  COEFFICIENT  OF  STABILITY,  OR  RATIO  OF  THE  ACTUAL 
TO  THE  ULTI]\rATE  THRUST,  WHEN  THE  CURVE  OF  PRES- 
SURE PASSES  THROUGH  THE  MIDDLE  OF  THE  KEY  AND 
THE   MIDDLE    OF   THE    JOINT    OF   GREATEST   THRUST. 

126.  Semicircular  arches  surcharged  liorizontaUy. — 
Table  DD.  The  ratio  ^,  is  seen  to  increase  from  1  oppo- 
site ^=1,  to  2.59  opposite  A^==1.35.  It  then  diminishes, 
and  would  finally  become  1  again. 

Opposite  A^=1.22,  h  is  1.91,  For  higher  values  of  K^ 
the  thrusts  given  in  that  table  were  calculated  on  the  sup- 
position of  a  curve  of  pressure  passing  through  the  middle 
of  the  key  and  the  middle  of  the  joint  60°  from  the  key. 
They  are  consequently  a  little  less  than  they  would  have 
been  had  we  continued  to  use  the  angle  of  maximum 
thrust.  ^ 

THICKNESS    OF   PIER UNIVERSAL    METHOD. 

128.  We  supjDose  the  surcharge,  if  partly  of  earth  or 
any  light  material,  to  be  reduced  in  height  in  the  jiropor- 
tion  of  its  density  comj^ared  with  the  density  of  the  arch 
proper.     Let 

/i=the  mean  height  of  the  pier  from  its  base  to  the  sur- 
face of  the  reduced  surcharge  over  its  top,  —E'  D,  fig. 
4;  =6>  0\  fig.  10;  —E'D\  fig.  15;  to  be  estimated 
if  not  known. 

E—\hQ  elevation  of  the  reduced  ridge  above  the  springing 
line,  ^Ca,  fig.  4 ;  =^CB:,  fig.  10;  ^QA,  fig.  9  ;  = 
m'i^',  fig.  15. 

^'=the  depth  of  the  arch  and  its  reduced  surcharge  at 
the  springing  line,  =Ga—E,  fig.  4 ;  —A  «',  figs.  10, 
11;  =77i' i?'=^,  fig.  15. 

?z=the  surface  of  that  part  of  the  arch  and  its  reduced 
load  which  lies  directly  over  the  half-span. 
12 


360 


THEORY    OF    THE    ARCH. 


?/?=tlie  moment  of  that  surface  in  relation  to  tlie  interior 

edge  of  tlie  joint  of  the  springing  line. 
Z=:the"lever  arm  of  the  thrnst  measni-ed  from  the  l)ase  to 

the  point  of  application  on  the  key-stone  joint,  say  J 

its  length  from  the  extrado?. 
i^=tlie  actual  horizontal  thrust  however  determined. 
'r=the  half-span,  whatever  he  the  curve  of  the  intrados. 
.T=the  distance  between  the  exterior  edge  of  the  base  of 

the  pier  and  the  intersection  of   this  base  with  the 

curve  of  pressure. 
6= the  nnknoAvn  thickness  of  pier. 

Let  us  determine  the  thick- 
ness of  pier  on  the  condition  that 
the  lowest  joint  shall  not  open 
at  its  inner  edge,  M.  We  must 
form  the  equation  of  moments  in 
relation  to  C3,  E'  c:i=^  E'  M— 
^€.     The  equation  is 


JS'    c^ 


Fig.  36. 


LeV,+ine+m=Fl ; 


(CG) 


(CT) 


If  we  desire  the  curve  of  pressure  to  intersect  the  base 
at  Ji5  from  the  exterior  edge  we  have  the  equation  of  mo- 
ments, 

^eViJrlne^-^n=El;  (66i) 

giving  


If  we  determine  the  thickness  of  pier,  as  in  the  method 
of  Audoy,  on  the  condition  that  tlie  curve  of  pressure 
shall  pass  through  the  exterior  edge  of  the  base,  we  have, 


UNIVERSAL    METHOD.  361 

lU^n€-^m=Fl;  (66^) 


giving 


If  tlie  curve  of  pressure  intersect  the  hase  at  |e  from 
its  exterior  edge,  we  liave  the  equation  of  moments, 

i^eVi^-^%ne-^in=Fl ;  (6G|) 

giving 

If  the  carve  of  pressure  intersect  the  base  at  any  pro- 
portional distance p%e  from  the  exterior  edge,  we  have 

{\-pyh-\-{l-p)ne^m^Fl;  (66;;) 

giving 

Let  the  curve  of  pressure  pass  through  the  middle  of 
the  base,  we  have 

ine+m  —  Fl:  (66^) 

e^J^^-'A  (674) 

\7)i      n  J 

The  values  of  ^,  drawn  from  (67),  (67^),  (67^),  (67t), 
.(67i),  (67^^),  differ  only  in  the  numerical  coefficients  of 

n    1'^     m      Fl 

so  that,  having  calculated  these  latter  quantities,  once  for 
all,  we  can  readily  solve  all  those  equations  and  others  of 
a  similar  character. 

Giving  to  h  and  I  in  (67)  the  particular  values  whioli 
correspond  to  the  springing  line,  we  learn  at  once  the 
required  thickness  of  the  arch  at  that  place,  on  the  con- 


362  THEORY    OF    THE    ARCH. 

tlition  that  the  curve  of  pressure  shall  cross  the  springing 
line  J  the  length  of  the  joint  from  the  exterior  edge. 

Sht;>uld  the  result  i-equire  an  increase  of  the  thickness 
already  adopted,  this  increase  need  not  extend  higher  up 
than,  say,  35°  from  the  si)ringing  line.  We  here  speak  of 
semicircular  and  semi-elliptical  arches. 

Giving  to  /  in  (6 7 5)  its  value  at  the  springing  line,  we 
learn  the  thickness  of  arch  required  at  that  place  when  the 
curve  of  pressure  passes  through  the  middle  of  the  joint. 


VALUES    OF    m    A^"D    11    IN    PARTICULAR    CASES  :    THE    KOOF   A 
PLANE   SURFACE,    THE    SEGMENTAL    KING   EXCEPTED. 

120.  TJie  intrado-s  a  semi-circle.     We  have  (see  art. 

65) 

7i33H^+^')-/-xO.T854. 

7y?  =  ?Xi^+i^')-?''X  0.452065. 
Kthe  roof  Le  inclined  45°,  we  have 

^/z:zzrX^/r-U.452065). 

130.  The  segmental  ring  of  equal  tldchness  tlwouglioiit 
(art.  72). 

,,=.|(7r^-l); 

??i=— — i/'XA"^— 1)(1— cos.  v)  ; 
in  which  v  is  the  semi-angle  at  the  center  and  s  the  span. 

131.  Segmental  arches  surcharged  horizontally. 

n  =  \s{f+d^t)-\r\^v-^m.  1v)  ; 

m  =  ls''{f-\-d^t)-r\],v^m.v+^QO^yv-\); 

in  which  /  is  the  rise,  d  tlie  thickness  of  the  arch  at  the 
key,  and  t  the  constant  depth  of  the  surcharge. 


ELLIPTICAL    ARCHES.  3G3 

132.  Segmental  arclies  covered  hy  two  symmetrical 
plain  surfaces  of  any  inclination  (art.  89)  ;  approximate 
formulge. 

133.  JEllijytical  arclies  surcliarged  horizontally  (art. 
107).    :E=E'=f+d-\-t; 

m^\i^  X E-i-^  X/X  0.452065  ; 

in  wliicli  T  is  tlie  lialf-span  or  semi-transverse  axis,  /  the 
rise  or  semi-conjugate  axis,  <^tlie  thickness  of  the  arch  at 
the  key,  t  the  constant  depth  of  surcharge  above  the  hori- 
zontal passing  through  the  extrados  at  the  key. 


THICKNESS    OF   PIER,    THE    ROOF     GREATLY   INCLINED,    LITTLE 
OR   NO    SURCHARGE. 

134.  When  the  roof  is  so  steep  and  the  arch  so  thin 
that  we  can  regard  the  triangle  D  E  P^  fig.  13,  as  forming 
a  part  of  the  semi-arch  or  pier,  we  can  give  to  w,  ???,  and 
7i,  in  the  formulae  of  art.  128,  a  meaning  which,  without 
changing  the  form  or  purport  of  any  of  those  equations, 
shall  render  their  application  somewhat  easier. 

Let  n—\hQ  surface  of  the  whole  semi-arch  and  its 
reduced  load,  and  of  that  part  of  the  pier  which  lies  above 
the  springing  line. 

m'=the  moment  of  that  surface  in  relation  to  the 
interior  edge  of  the  joint  of  the  springing  line. 

li—\k^  height  of  the  pier  from  its  base  to  the  springing 
line. 

Let  E,  Z,  F^  r,  a?,  e  remain  the  same  as  in  art.  128. 

Let  /,  as  usual  represent  the  angle  between  the  roof 
and  a  vertical.     We  have 


364  THEORY     OF    THE    ARCH. 

n'  =  ^  tang,  /x  B'-r'  X  0.7854. 

7u'=^  tang.  IxE-{r-^  tang.  7x^)-?-'X  0.452065  ; 
To  determine  the  tliicknes.s  of  jiier  -we  have,  for 

x='^;  yit^+ine^m'=Fl:  (66)' 

:c={e;  \h't'+^ne^m=Fl,  (66-]-)' 


,=.-1X^+^/1X^,-4^+4^,      (6.1) 
.r=0;  i7/e^+/i'e+?//  =  i^/,  (66^)' 


.r=|^;  tVA'^-KV»'«+"'=^'.  (661)' 


,r=i<';  i«'(;+;//=F?,  (66i) 

.^o/^.-')  (670' 

To  (67)  and  (07)'  we  should  generally  look  for  the 
thickness  of  pier. 


THICKNESS    OF     PIER.— i;XAMPLES. 


365 


THE   THICKNESS    OF   PIER — EXAilPLES. 

135.  Example  I.     Powder  magazine,  roof  inclined  45°, 
no  surcharge.     r=:10';  i?=12';    7r=^=1.20;    7=45°; 

7,':zrlO';  Z=A'^/'+|(^-r)  =  21'.33?)3..;  F,  table  FF,=: 
r'x0.3T90  =  37.90;  n\  art.  134,=65.46;  m',  art.  134,= 
173.356  ;  from  these  data  we  obtain 


=80.8533, 


-|,=6.546;  |-2=42.85;  ?^=1V.3356  ; 

and,  by  (67)'  art.  134,  f=10'.41. 

The  rule  of  Audoy,  art.  67^  exam- 
ple 1,  gives  6=8'.79. 

For  /i'=5'  (67)'  gives  ^=:8'.75  ; 
the  rule  of  Audoy  gives  for  A =5',  e— 
8M9. 

For  A=0,  (66)'  gives  e=5'.87 ; 
the  rule  of  Audoy  gives  e=6'.81. 

From  the  value  of  e  at  the  sj^ring- 
ins:  line,  5'.87,  we  learn  that  the  curve 
of  pressure  passes  through  <?2  at  the 
distance  |6^=:3'.91  from  A^  the  in--'- 
terior  edge  of  the  joint.  This  is  en- 
tirely outside  of  the  arch  proper,  supposed  to  be  of  equal 
thickness  throughout.  Hence,  making  A  0=:5'.S1^  the 
arch  must  be  extended  to  the  line  0  0\  either  straight  or 
slightly  convex  on  the  upper  side,  joining  the  i)roper 
extrados  about  40°  from  the  spriuging  line,  where  the  curve 
of  pressure  will  obviously  pass  near  the  middle  of  the 
joint.  The  point  ci  where  this  curve  comes  nearest  to  the 
intrados,  is,  table  FF,  about  37^°  from  the  vertical.  In 
practice,  it  would  be  better  to  make  the  whole  lower  part 
of  the  arch,  out  as  far  as  the  roof,  one  solid  piece  of 
masonry. 

Example  2.  The  mas^azine  of  Fort  Jefferson,  fig.  12. 
r  =  U'',     F.  =  ir.50;     7f=:1.25;     7=56°  3'  23"=55°+ 


Fig.  3T. 


366  THEORY     OF    THE    ARCH. 

1°.0564;  ^=5'.90;  Ji  =  lQ'.50  Lelow  the  springing  line-f 
13'.50  above,=30';     /:=10'.o04-U'-|X  ;V.50==32'.S333. 

i:=-^-^+t=:2r;  ^=^-/'C0taug.  /=U'.5S;    i<^*= 
sin./  '  o  ' 

114.7128;  n,  art.  li?0,  =  158.12 ;    ?/?,  art.   129,  =  1097.82 ; 

^.=5.27;  ■^,-2tlS;  ^=36.594;  4^=125.5468;    ^  = 

6.943.     With  these  data  we  obtain : 

(67),  art.  128,  ^=J^,    .^      .         .         .         .         f=14'.85 

The  rule  of  Amloy  gave' art.  67,  ex.  2,     .         .     <?=ir.88 
To  obtain  the  reciuired  thickness  at  the  spring- 
ing line,  make  A=13'.50;  /=16'.3333;  and 
we  have,  (67),    ......       6=6'.47 

The  rule  of  Audoy  give.s,      ....  ^=7.33 

Example  3.  Arch  in  the  citadel  of  Fort  Porter,  fig.  11, 
art.  63,  ex.  3,  art.  67,  ex.  3.  rz=zQ,' ;  i?=7'.668,  A^=:1.278  ; 
7=86°  46';  ^=6'.60 ;  //  =  18'  below  the  springing  line-f- 
12'  above,  =  30';  ?=18'+ r +§(7?-r)  =  25'.n2  ;  J^= 
14'.28;  ^'=12'.59;  71=52.3350;  7^^  =  149.254. 

This  case  does  not  come  directly  under  table  FF;  but, 
comparing  that  with  table  DDD,  we  infer  that  the  coeffi- 
cient of  stability  is  about  1.69  ;  the  ultimate  thrust  of  the 
arcli,  Avithout   including  the   effect  of  the  surcharge   C/f, 


•  TaUe  FF,  A'=  1.2.5,  7=55',           .         .         .         i^=r^x  0.2916     0.2'.tlG 
Table  FF,  A'=  1.25,  7=60° 7'"=  r»  x  0.2529 


Difference  for  tlie  interval  of  5'  in  7, .  =r''x  0.0387 

5°  :  0.0387  ::  r.0564  :  «=  ....         0.0082 


Hence  the  thrust  of  the  arch  •without  surcharge,  .         .     ^r'x  0.2834=55.5464 

AJ«1  for  gurchnrge,  last  column  of  table  FF,  .4  =  14' x  5.90  x0.7163         .     =59.1664 


Total  thru3t=7''=114.7128 
Tlie  nltiniate  lliruM,  art.  67,  ox.  2,  is  74.80  ;  the  coeflBcient  of  stability, 

table  FF,  A'=  1.25,  7=55',  60%  i.s  1.53.     74.86x1.53       .         .         .     =114.5358 


THICKNESS    OF    PIER.— EXAMPLES.  367 

deep,  art.  63,  ex.  3,  is  5.0126,  wliicli,  multiplied  by  1.60, 

gives, 8.4713 

Add  for  surcharge,  table  FF,  A=rtC=^'  X  G'.6  X 

0.705, =27.9180 


Total  thrust,  i^=  36.3893 
^  =  1.7445;  -^2=3.0435;  -^=4.975;  -y-=r30.4603 ;  — = 

lb  it  Ih  IL  )li 

2.852. 

With  these  values  we  obtain  from  (67)  — a;=^<?,     6=9'.36 

The  rule  of  Audoy  gave,  art.  67,        .         .         .     ^=6'.65 

To  obtain  the  required  thickness  at  the  springing  line, 
make  7^=12' ;  ?=7'.112,  we  find,  .  .  .  ^=2'.72 
The  rule  of  Audoy  gives  at  the  springing  line,         <?=2.85 

By  comparing  tables  FF  and  DDD,  we  find  that  the 
angle  of  greatest  thrust  of  the  arch  without  surcharge,  is 
about    .........         55° 

The  angle  of  the  maximum  effect  of  the  surcharge, 

table  FF,  is 48° 

Taking  both  into  view,  the  angle  of  greatest  thrust  is, 

therefore,  about 50° 

"We  now  have  three  points  of  the  curve  of  pressure  in  the 
arch,  one  at  the  key  J  the  length  of  the  joint  below  the 
extrados ;  one  at  50°  from  the  key,  the  same  distance  from 
the  intrados ;  and  one  at  the  springing  line  f  X2'.72  =  l'.82 
from  the  intrados.  Between  the  first  and  the  second,  the 
curve  is  near  the  middle  of  the  joints  ;  below  the  second, 
it  gradually  departs  from  the  intrados  and  passes  through 
the  middle  of  some  joint  about  60°  from  the  key,  and  at  the 
springing  lines  passes  entirely  outside  of  the  arch  proper ; 
hence  the  necessity  of  enlarging  the  arch  at  that  line,  as  ftir 
as  e=2'.72,  or  a  little  further,  the  enlargement  diminishing 
as  we  ascend,  and  becoming  nothing  about  60°  or  65°  from 
the  key. 

Example  4.  The  magazine  of  Yauban,  fig.  13,  arts.  63, 


368  THEORY     OF    THE    ARCH. 

67.    r=12'.50:    i?=15'.50;    7i^=1.24  ;    7=40°  7' 17"= 
50°-r/.S786;    Ji'=S';    Z=A'+/-+§(7.^-r)  =  22'.50 ;  ^= 
20.50  ;  ?i\  art.  134,  =  120.039  ;  m\  art.  134,=235.06. 
Table  FF,  jr=1.24, 1= 

45°,         .         .         7^=:?-2x  0.4076 
Table  FF,  A  =  1.24,  7= 

50';       .        .         7  =  /-'x0.3372  =  /--x0.3372 


Difterence  for  the  inter- 
val 5°  in  7,  .         ^-^X  0.0704 
5° : 0.0704  ::0°.8786:.r=  .         .     ?'-X 0.0124 


Therefore,  for  7r=1.24, 

7=49' 7' 17"     .         .         .        7^=^^X0.3496=54.625 

,/  7/2  ^'  Wl 

-^,=15.005;  yr2=225.1463  ;  -^^=29.3825;  -y7-=153.633. 

li  It  lb  Ih 

Substituting  these  values  in  (67)',  art.  134,  we 

find f=10'.56 

The  rule  of  Audoy  gave  .....     e^=  9'. 2 3 
To  obtain  the  required  thickness  at  the  springing  line, 

make  //'=0,  and  Z=14'.50  in  (66)'  art.  134,  we  find  6=6'.96 

The  rule  of  Audoy  gives,  under  the  same  circum- 
stances,        .......     6= 7'. 30 

Example  5.  Chester  Bridge,  Harrison,  architect : 
Span = •5  =  2 00';  rise=/=42';  7=  thickness  at  the  key= 
4' ;  7'=thickness  at  the  spring=6' ;  r=radius  of  the  intra- 

dos=140',  nearly;  6  =  4.76 X/;    7^=1 +  -=1.0286.     We 

have  no  certain  information  as  to  the  load  borne  by  this  arch ; 
it  had  necessarily  some  surcharge  at  the  crown  ;  let  us  sup- 
pose its  cover  of  earth  and  masonry  to  have  the  density  of 
the  arch,  and  to  rise  to  a  horizontal  i)lane  one  foot  above 
the  extrados  at  the  key,  /=!'.  We  obtain  the  thrust,  by 
proportional  pai'ts,  from  table  EE. 


CHESTER    BRIDGE.  869 

0.0622   .    .  0.0622    O.OTOO   .    .  O.OYOO 
0.0529  0.0611 


1  :  0.0093  : :  0.T6  :  ;r=0.00n  1 :  0.0089  : :  0.Y6  :  a?=0.0068 


0.0551  0.0632 

0.0632 
0.0551         .         .     0.0551 


1  :  0.0081  : :  0.86  :  a?r=0.0070 


Tlinist  of  the  arch  proper,      .     ?'"X  0.0621  =  1217.16 
Effect  of  surcharge,       ^'^(7=140' Xl'X  0.88=   123.20 

Total  thrust,  i^=1340.36 

This  thrust  corresponds  to  a  curve  of  pressure  at  the 
key,  ^d  froui  the  extrados,  and,  at  the  spring,  the  same 
distance  from  the  iutrados.  But  in  consideration  of  the 
lightness  of  the  arch,  we  ought  to  provide  for  a  larger  thrust, 
corresponding  to  a  curve  of  pressure  passing  through  the 
middle  of  the  key,  and  at  least  2'=^^d'  from  the  intrados 
at  the  spring. 

"We  learn  from  table  EE,  that  the  actual  thrust,  when 
^— 4.76x/,  ^=1.0286,  exceeds  the  ultimate  thrust  by  a 
little  more  than  9  per  cent.  It  is  obvious  that  if  we  add 
4|-  per  cent,  to  the  ultimate,  or  in  this  case,  about  4  per 
cent,  to  the  actual  thrust  above  given,  we  shall  have  nearly 
the  thrust  corresponding  to  a  curve  of  pressure  passing 
through  the  middle  of  the  joints,  here  supposed  to  be 
equal. 

Thus,  1340.36 
+  4  per  cent.,       53.61 

1393.97 
In  fact  the  actual   thrust,  according  to 

this  last  supposition,  art.  116,  ^'=:45°, 

is 1377.00,  nearly. 

The  true  semi -angle  at  the  center,  is  t'=45°  35'  5". 


370  THEORY    OF    THE    ARCU. 

Let  us  therefore  take,  as  tlie  total  thrust,  i<'=1400. 

The  mean  pressure  per  square  foot  at  the  key  is  J-^V"" 
=  350  cubic  feet;  and  the  pressure  at  the  upper  or  most 
exposed  edge  we  must  regard  as  twice  this  mean  pressure, 
or  700  per  square  foot.  If  we  suppose  the  material  to  be 
granite  weighing  175  pounds  per  square  foot,  this  last  pres- 
sure corresponds  to  about  850  pounds  per  S(piare  inch,  not 
more,  probably,  than  yV  the  ultimate  resistmg  power  of 
good  granite. 

The  pressure  at  the  springing  line  being  the  resultant 
of  the  thrust,  F,  and  of  the  Aveight  of  the  semi-arch  and  its 
load,  7^,  determined  below,  is 

P=/(1400/H-(1802)''=2282  cubic  feet; 

which  gives  the  pressure  at  the  intra<los  of  the  springing 
line 

0  X—— =  7601  cubic  feet=924  pounds  per  square  inch. 
6 

Suppose  we  wish  to  obtain  the  thickness  of  an  abut- 
ment pier  23'  high  from  the  foundation  to  the  springing 
line.  This  gives  7^=23+47  =  70' ;  /=67'.666  ;  n,  art.  130, 
=  1802;  777,  art.  130,  =56256.     Hence,  by  calculation, 

|=25'.74;  p=662.70;  ^=803.66;  ^-=1353.33. 

These  quantities  give  us,  art.  128, 

(67),  when  x^\e,    6=25'.65. 
(67i)     "     x^\e,    ^=22'.13. 
At  the  springing  line,  Z=  44.666,  and  the  thickness 
when  x—\e,  or  when   the  resultant  passes  through   the 

middle  of  the  base,  is  ^=2  X — - —  =  6  .97. 

Example  6.  Central  arch  of  the  London  Bridge.  El- 
liptical. r=half-span  =  76' ;  /=rise=38' ;  f7=thickness  at 
the  key =5'. 


LONDON    BRIDGE. 


6  i 


Suppose  the  load  of  the  density  of  masonry  to  rise  to 
the  level  of  the  top  of  the  arch,  and  the  curve  of  pressure 
from  the  key  to  the  joint  of  greatest  tlirust  to  continue 
near  the  middle  of  the  joints.  The  thrust  already  given 
in  art.  121,  is  i^'^z-'x  0.16723=966.45  ;  to  this,  if  we  sup- 
pose a  surcharge  of  1',  we  must  add  ^  =  r^x  1.0617  = 
161.38  ;  giving,  as  the  total  thrust,  i^=  1127.83. 

Let  us  assume  as  the  point  of  apj^lication  of  the  thrust, 
the  center  of  the  keystone  joint,  2^  feet  above  the  intra- 
dos  ;  let  us  also  assume,  as  the  height  of  the  pier  from  its 
base  to  the  springing  line,  16'.  We  have  A=:16'+38'  +  5' 
+1'=60';  ?=56'.50;  U=U'=U'',  art.  133,  w==1075.76, 
w=27849; 


n 


n 


in 


Fxl 


^=17.93;  ^=321.46;  ^=464.15;  -^-'=1062.04. 

These  values  substituted  in  (67),  a?=J<?,  give  e=:33'.95. 

(67i),  ^■=1^,     "     6=2S'.92. 

(67^),  a?=0,  6=2r.02. 

At  the  springing  line  we  have  A=44' ;  Z=40'.50 ; 

giving,  when  x=^e^     .  .  .  ^=20'.54. 

The  true  extrados  of  the  arch  should  extend  to  IJ ; 
A£=20'54. 

From  the  key  to  o  n,  0^:)= about  ^a  Cj  the  thickness  of 
the  arch  may  remain 
nearly  constant:  from 
0  n  the  thickness  should 
gradually  increase  to 
AB^  or,  what  is  still 
better,  the  lower  part 
of  the  arch  should  form 
one  solid  mass  with  the 
pier  up  to  some  joint 
t  <9,  more  or  less  elevated,  according  to  circumstances,  mak- 
ing t  s  the  true  springing  line. 


Fio.  8S. 


oi'2  tueort   of  the  arch. 

thickn:ess  of  auch. 

136.  The  celelirated  Perronot  has  driven  the  followinc: 
rule  furtlie  thickness, <'/,  at  the  key  in  terms  of  2/'  the  span: 

(7=l'.l"+3-VX2;"=13  inches+^V  the  span. 

lie  does  not  seem,  however,  to  have  paid  mnch  atten- 
tion to  the  rule ;  but  has  made  his  bridges  much  lighter 
than  the  rule  would  require. 

The  best  information  on  this  subject  may  be  obtained 
from  the  record  of  existing  structures  which  have  stood 
the  test  of  time.  Table  I  gives  the  principal  elements  of 
many  celebrated  bridges,  differing  widely  in  their  absolute 
and  relative  dimensions.  Numbers  4,  9,  and  19  are  re- 
markable for  their  lightness ;  but  No.  4  fell  on  the  removal 
of  the  center,  and  it  is  pi-obable  that  the  other  two  are 
near  the  limits  of  possiljility.  The  segmental  bridges  of 
Rennie  and  Stevenson  are  somewhat  heavier,  but  still  light. 
They  are  admiral)le  works,  and  show  what  may  be  done 
with  good  stone. 

In  these  lighter  structures,  which  may  be  regarded  as 
models  of  segmental  arches,  the  thickness  at  the  key  is 
found  to  vary  from  -3'^  to  3^3-  the  span,  and  from  ^'^  to  -3'^ 
the  radius  of  the  intrados.  The  augmentation  of  thickness 
at  the  springing  line  is  made,  by  the  Stevensons,  from  20 
to  30  per  cent.;  by  the  Kennies,  about  100  per  cent. 

The  London  Bridge,  in  its  plan  and  workmanship,  is, 
perhaps,  the  most  perfect  woi-k  of  the  kind.  The  intrados 
is  an  ellipse  ;  the  span,  152';  the  rise,  |  as  much,  the  thick- 
ness at  the  key  -3^  the  span.  The  crown  settled  down 
only  two  inches  on  the  removal  of  the  center. 

The  proper  thickness  at  the  key  does  not  depend  upon 
the  rise  and  span  alone,  but  also  upon  the  load,  and  upon 
the  resisting  power  of  the  material. 

The  pressure  at  the  extrados  of  the  key,  which  is  in 
general  the  most  exposed  part  of  that  joint,  should  not, 
according  to  the  best  authorities,  exceed  j\  tlie  ultimate 


THICKNESS    OF    ARCH.  3*73 

resisting  power  of  tlie  material.  This  rule  is,  without 
doubt,  perfectly  safe.  In  the  railway  bridge  of  Maiden- 
head, No.  13,  table  I,  the  ultimate  resistance  is  only  3j 
times  the  calculated  actual  pressure  at  the  extrados  of  the 
key. 

Although  we  may  determine  the  thrust  on  the  suppo- 
sition of  a  curve  of  pressure  2:)assing  through  the  middle  of 
the  key-stone  joint,  we  ought  to  admit  the  possibility  of  a 
greater  elevation  ;  viz. :  to  ^  the  length  of  that  joint  from 
the  extrados.  This  diminishes  the  thrust ;  but  the  differ- 
ence in  light  arches  is  very  small. 

Let  P=the  pressure  per  unit  of  surface  at  the  upper 
edge  of  the  key-stone  joint ;  ^=:the  actual  thrust,  which, 
for  our  present  purpose,  we  may  regard  as  acting  -i-  the 
length  of  the  joint  from  the  extrados ;  r/==:the  length  of 
the  key-stone  joint.     We  have,  art.  112, 

p  —  9 

F . 

-J  is  the  mean  pressure ;  which,  to  state  the  rule  in  another 

way,  should  not  exceed  -^-^  the  ultimate  resistance  of  the 
material. 

By  two  or  three  trials  we  can  always  find  a  value  of  d 
which  shall  satisfy  the  condition. 

Being  once  in  its  neighborhood,  we  can  determine  that 
value  directly  by  an  equation  of  the  first  degree. 

But  it  is  far  more  satisfactory  to  make  two  or  three 
independent  trials,  and  to  observe  the  resulting  variations 
in  the  thrust  and  in  the  pressure  per  unit  of  surface. 

The  load  of  a  bridge  is  generally  fixed  by  circum- 
stances, and  independent,  or  nearly  so,  of  the  thickness  of 
the  arch. 

Example.  The  Monocacy  stone  bridge,  art.  73.  6-= 
span=:54';  y=rise==:9';  r=:radius  of   the    intrados=:45'. 

Suppose  the  load  of  this  aqueduct  bridge  to  rise  to  a 
plane  8'  above  the  crown  of  the  arch,  and  to  have  the 
density  of  the  latter  ;  ^=8'. 


374:  THEOKY     OF    THE    ARCH. 

Assume  4,000  pounds  per  square  iucli  as  the  ultimate 
resisting  power  of  the  material  =  576,000  pounds  ]^er 
square  toot =7^'.  Assume  tbe  weight  of  a  cuLic  foot  of 
matei-ials  to  be  160  pounds. 

First,  in  table  EE,  under  <s=6/,  make  7^=1.05.  We 
have 

F     ?-2x0.0695+rXifX0.8476     _^,_      ,.    ^ 
¥= ^^<T0~5 =198.166  cubic  feet; 

rp 

which,  multiplied  by  160,  gives     .         -^=31706  pounds. 

But  i-  P'=  .  .  .  28800       " 

Consequently  d  must  be  increased.     Make  7^=1.06. 
-pp 

We  now  have  -y=27082  pounds.      Hence,  by  propor- 

jp 

tional  parts,  the  value  of  K  which  shall  make  -y=:28800 

pounds=:yV^',  is  7r=1.0563,  giving  r7=r2'.53. 

In  making  the  above  calculations,  we  have,  without 
doubt,   under-estimated  the  sti-ength  of  the  stone.     The 

F 

mean  pressure,  -r-,  corresponding  to  7=r2'.53,  is  probably 

not  more  than  -^-^;P'. 

Some  excess  of  strength  at  the  key  may  be  necessary 
to  keep  the  pressure  per  unit  of  surface  at  the  springing 
line  or  weakest  joint,  M'ithin  the  limits  of  safety. 

Our  various  tables  of  the  actual  thrust  furnish  the 
means  of  solving  most  questions  of  this  kind  in  a  few  min- 
utes. But  we  f>uglit  not  to  aim  at  great  accui-acy  of 
adjustment.  The  resisting  power  of  stone  is  imperfectly 
knowm. 

Very  light  arches  may  seem  to  stand  the  test  above 
given,  and  still  be  impracticable :  that  is,  the  curve  of 
pressure,  in  certain  places,  may  necessarily  pass  outside  of 
the  prescribed  limits  ;  and  the  pressure  per  unit  of  surface, 
at  the  most  exposed  edges,  may  be  far  more  than  double 
the  mean  pressure. 


INCREASE    OF    THICKNESS. 


37J 


Tliis  will  he  explained  hereafter :  see  Third  mode  of 
rupture,  Limit  of  practicaLle  arclies,  &e. 


INCREASE  OF  THICKNESS  AT  THE  REIN'S,  OR  j;omT  OF  GREAT- 
EST THRUST,  AND  BELOW  THAT  JOINT  TO  THE  SPRINGING 
LINE. 

137.  It  will  be  shown  hereafter  that  very  light  arches, 
otherwise  impracticable,  may  be  made  practicable,  l)y  giv- 
ing a  certain  increase  of  thickness  between  the  key  and  the 
joint  of  greatest  thrust,  without  changing  the  thickness  at 
either  of  those  joints.  In  this  article  we  shall  have  in  view 
larger  arches,  not  in  danger  of  the  third  mode  of  rupture. 

Semicircular  arches.  We  learn  from  table  DDD,  that 
the  joint  of  greatest  thrust  is  about  the  same  in  circular 
arches  of  the  ordinary  proportions,  under  the  three  suppo- 
sitions there  made  in  reference  to  the  curve  of  pressure. 

1st.  The  curve  of  pressure  pass- 
ing through  I  and  m,  as  in 
the  ultimate  thrust. 

2d.  Through  c  and  C2,  ch—^ah^ 
mG2=\mn^  as  supposed  in 
table  DDD. 

3d.  Through  c  and  cs,  ch=lah, 
mc2=z^m7i^  as  in  table  DD. 

We  learn,  furthermore,  that 
the  joint  in  c^uestion  is  in  the  neighborhood  of  60°  from 
the  key. 

We  also  know  from  observation  that  the  circular  arch, 
if  we  except  those  of  very  light  and  impracticable  propor- 
tions, tends  to  open  at  the  intrados  of  the  key  and  at  the 
extrados  of  the  reins.  We  therefore  know  that  the  curve 
of  pressure  is  above  the  middle  of  a  b,  and  inside  the  mid- 
dle of  m  11. 

Any  increase  of  thickness  between  a  h  and  m  n,  would 
13 


3TG 


THEORY    OF    THE    ARCn, 


be  useless,  unless  the   ai'cli  be  exceedingly  light,  so  as  to 
ehanore  its  form  hy  vibration  under  a  variable  load.     Such 
increase  will  have  no  sensible  effect  upon  the  thrust ;  it  will 
cause  no  sensible  change  in  the  place  of  the  curve  of  pres- 
sure ;  it  will  not  diminish,  l)ut  increase,  the  pressure  per  unit 
of  surface  at  m,  the  edge  most  exposed.     For  all  these  rea- 
sons it  will  be  useless.     But  the  curve  of  pressure,  after 
crossing  the  weakest  joint,  at  <?2,  begins  to  approach  the 
extrados ;  some  degrees  below  7n  n  it  crosses  the  central 
line  of  the  joints ;  it  soon  passes  the  exterior  limit,  and 
usually  runs  entirely  outside  of  the  circular  ring,  at  some  dis- 
tance above  the  springing  line.    It  will  be  enough,  therefore, 
to  begin  to  increase  the  arch  at  some  joint,  t  S,  about  25° 
above  the  sjiringing  line.     Find  the  thickness,  BA,  at  the 
springing  line,  art.  128,  (67),  on  the  condition  that  Bc3  = 
\B  A,  or  x=^e.     B  S  will  be  the  proper  extrados  of  this 
j)art  of  the  arch.     For  greater  security  we  may  extend 
this  line  out  a  little  further  than  B  S  a^  above  given. 

138.  Elliptical  arclies.  We  have  found  the  actual 
thrust  of  the  elliptical  arch  to  be  about  equal  to  that  of 
a  circular  arch  of 
the  same  span,  and 
of  a  thickness  at 
the  key  and  a  thick- 
ness of  load  as  much 
larger  tlian  the  cor- 
responding pai-ts  of 
the  elliptical  arch 
a.s  the  half-span  is 
larger  than  the  rise. 
Calling  2/'  the 
span, /the  rise,  and 
d  the  thickness  at 
the  key,  we  have, 
as  the  thickness  of  p^^  ^ 


INCREASE    OF    THICKNESS. 


377 


tlie  auxiliary  circular  iirc\  B  A^^—id.  With  one  excep- 
tion, tlie  pressure  per  unit  of  surface,  the  elements  of  sta- 
bility are  the  same  in  the  two  arches ;  their  respective 
curves  of  pressure  have  the  same  relative  situations,  and 
these  curves  finally  meet  and  cross  at  tlie  springing  line. 
The  true  extrados  should  be  another  ellipse,  on  the  axes 
£  O,  Ch^  similar  to  the  intrados.  But  near  the  springing 
line  the  thickness  must  be  increased. 

Determine  the  thickness  required  at  the  springing  line 
in  order  that  Dc-i—\D  A,  or  x—\e,  art.  128,  (67). 

The  tangent,  D  S^  will  be  the  proper  extrados  of  the 
lower  part  of  the  arch,  or  some  curve  near  that  tangent. 
This  extrados,  of  course,  cannot  extend  beyond  the  middle 
of  the  pier  when  the  latter  supports  parts  of  two  arches. 

139.  Segmental  arches.  Wlierever  we  suppose  the 
curve  of  pressure  to  be,  the  joint  of  greatest  thrust  is  at 
the  springing  line,  or  but  little  above  it.  If  the  arch  be 
of  sufficient  weight  and  size  to  resist  all  change  of  form, 
its  thickness  may  remain  the  same  throughout.  The  idea 
of  increasing  the  thickness  in  order  to  equalize  the  pres- 
sure per  unit  of  surface  at  the  key  and  spring  is  altogether 
fallacious. 

All  observation  show^s  that  the  segmental,  like  the 
circular  arch,  tends  to  open  at  the  intrados  of  the  key 
and  the  extrados  of  the  joint  of  gi'eatest  thrust.  Conse- 
quently, the  curve  of  pressure 
starts  above  the  center  of  ah^ 
and  ends  in  the  arch,  within  the 
center  of  771  nz=za  h. 

Let  us  first  suppose  m  c  2  = 
i7?27^=:i-(/,  and  the  joint  m  n  in- 
creased to  m  r.  This  increase 
having  no  sensible  effect  upon 
the  thrust,  the  entire  pressure  is  fig.  41. 


378  THEORY    OF    THE    ARCH. 

still  borne  by  m  n  ;  the  extension,  n  r,  remains  open  ;  the 
pressure  at  n  remains  zero ;  at  m,  as  before,  it  is  double 

the  entire  pressure,  P^  divided  l)y  mn^  =  — . 

Let  us  now  suppose  the  curve  of  pressure  to  pass 
through  the  middle  of  m ;?,  m  C2=:^?/i  ?i  :  the  pressure  per 

unit  of  surface  is  everywhere  -^ ;   nicrea.se  the  jomt  ?7i  n 

to  mr:=^\inn.     The   pressure  at  r  is  zero;   at  m  it  is 

— --=-x— r.    Thus,  increasinsr  the  ioint  by  one  half,  we 

have  increased  the  pressure  per  unit  of  surface  at  ?;?,  by 
one  third. 

Supposing  the  arch  to  be  of  e(|ual  thickness  through- 
out, so  long  as  the  curve  of  pressure  meets  the  springing 
line  inside  of  its  middle  point,  any  increase  of  the  joints 
would  be  worse  than  useless. 

There  are,  however,  other  considerations  which,  in  the 
case  of  veiy  light  arches,  demand  a  progressive  augmenta- 
tion of  thickness  from  the  key  to  the  spring.  We  allude 
to  stiffness,  stability  of  form  under  variable  loads.  No 
precise  rule  can  be  given,  but  it  can  hardly  ever  be  neces- 
sary to  carry  the  augmentation  beyond  fifty  per  cent. 
This  augmentation  will  c:ive  a  wide  rano-e  to  the  curve  of 
pressure,  within  the  limits  of  perfect  safety  to  the  upper 
and  lower  edges  of  the  lower  joint. 

Such  augmentation  will,  it  is  true,  always  increase  the 
pressure  per  unit  of  surface  at  m^  the  intrados  of  the 
weakest  joint,  whatever  be  the  curve  of  the  intrados;  but 
it  will  diminish  the  pressure  at  the  exterior  edge  11  or  r; 
and  it  may  prevent  the  third  mode  of  rupture. 

The  most  important  principle  to  bear  in  mind  here,  is 
this :  if  the  pressure  |)er  unit  of  surface  at  in  is  too  great, 
we  must  increase  the  thickness  of  the  segmental  ring 
throughout. 


RELATIVE    PRESSURES.  37 


RELATIVE   PRESSURE,    PER     UNIT   OF   SURFACE,    AT    THE   KEY, 
AND    THE   WEAKEST   JOINT   BELOW   THE   KEY. 

140.  The  pressure,  F^^  on  any  joint  m  n^  is,  F^  — 
|/^2_|_^^2^  in  wliicli  expression  -F' represents  the  horizontal 
thnist,  and  n  the  surface  of  the  arch  and  its  load  between 
the  joint  m  n  and  the  summit. 

In  flat  arches  n  is  less  than  F^  and  F^^,  therefore,  less 
than  1.414 xi^;  if  n—F,  we  have  F^—\A\^y.F;  if 
n  =  \n^%F,  or  n^=:?>F\  we  have  F^—lF. 

It  is  easy  in  any  arch,  to  determine  the  pressure  upon 
every  joint ;  but  it  is  only  in  reference  to  the  weakest  joint 
that  this  knowledge  is  important. 


PRESSURE,  PER   UNIT    OF   SURFACE,  UPON    THE    LOWEST   JOINT 

OF   THE   PIER. 

141.  Referring  to  the  notation  of  arts.  112,  128,  the 
pressure  P^  per  unit  of  surface,  at  the  exterior  edge  of  the 
base,  if  a?=J^,  is 

P— 2x^i^ 

e 

Should  this  exceed  yV  the  ultimate  strength  of  the 
material,  it  will  be  necessary  to  increase  e. 

The  above  refers  to  an  abutment.  The  pressure,  per 
unit  of  surface,  upon  the  base  of  a  pier  supporting  two 
equal  arches,  is 


380 


THEORY    OF    THE    ARCH, 


Let  6'=tlie  tbickuess  required  at  the 
si:)rmging  line  to  support  safely  the  given 
pressure  ;  7<'=:the  height  of  ])ier  above  the 
sprinpng  line  ;  //=the  height  below  ;  6= 
the  thickness  at  bottom. 

Let  us  determine  6  so  as  to  equalize  the 
pressure,  per  unit  of  surface,  upon  e'  and  e; 
we  must  have 

^n^e'l/ _1n-\-eh'-\-\h{e'+e)  . 


Fig.  42. 


gr 


THIRD    MODE    OF   RUPTURE LBIIT   THICKNESS    OF   TOSSIBLE 

AKCIIES. 

142.  In  article  25,  the  conditions  of  this  mode  of  rup- 
ture have  l^een  briefly  pointed  out. 

The  liglit  semicircular  arch  surcharged  horizontally  up 
to  the  summit  of  the  crown,  and  without  additional  sur- 
charge, is  the  one  most  exposed  to  this  danger. 


Fig.  48. 


In  this  rupture,  the  arch  rises  at  the  crown  and  falls  in 
four  segments  ;  the  upper  segments  turn  outwardly  on  the 
hinsres  7i\  n  ;  the  lower  segments  fall  towards  the  center, 
turning  on  the  hinges  ???,  m.  The  work  below  m  ;?,  mn., 
is  not  necessarily  disturbed.     At  the  moment  of  equili- 


LIMIT    OF    POSSIBLE    ARCI1E3.  381 

brlum  preceding  tlie  rupture  and  fall,  the  points  or  edges 
in  contact  are  a,  n\  n\  m,  m.  The  curve  of  pressure, 
which  necessarily  passes  through  those  points,  is  tangent 
to  the  intrados  at  «,  m,  w,  and  to  the  extrados  at  n\  n'. 
The  joints  m  ??,  on  n  are  the  lower  joints  of  rupture  and 
the  joints  of  maximum  thrust,  supposing  a  to  be  the  point 
of  application  of  the  horizontal  thrust. 

This  rupture  can  only  take  place  in  light  arches  ;  ana 
in  these  the  position  of  m  n  is  nearly  the  same  as  in  the 
first  mode  of  rupture — ultimate  thrust. 

Let  us  determine  the  limit  of  possible  arches,  suppos- 
ing, for  the  present,  that  mere  edges  of  masonry  are  able 
to  withstand  any  pressure  whatever. 

Let  i^=the  maximum  thrust  of  the  arch,  or  horizontal 
force  which,  applied  at  «,  shall  keep  the  segment  I  a  m  n  r  I 
in  equilibrium  on  m^  m  n  being  that  particular  joint  which 
renders  F  a  maximum. 

Let  i<"=the  horizontal  force  which,  applied  at  a^  shall 
be  just  sufficient  to  overturn  the  upper  segment  of  the 
arch  on  some  point  n  of  the  extrados.  It  is  evident  that 
F^  in  any  arch,  must  not  exceed  F.  When  these  two 
forces  are  equal,  we  shall  have  the  thinnest  possible  arch, 
an  arch  in  which  the  curve  of  pressure,  as  already  stated, 
touchino^  the  intrados  at  a  and  w,  is  in  contact  with  the  ex- 
trados  at  ?i'.  In  any  thinner  arch,  the  curve  of  pressure 
would  necessarily  pass  outside  of  the  ring,  either  at  a  or  n 
or  both.  In  other  words,  we  are  not  at  liberty  to  suppose 
a  thinner  arch. 

It  is  interesting  to  observe  that  this  kind  of  rupture 
may  take  place  without  any  disturbance  of  the  pier  or 
lower  part  of  the  arch — all  motion  being  confined  to  that 
part  of  the  arch  which  lies  above  the  lower  and  regular 
joint  of  rupture,  m  n.  On  the  other  hand,  the  fii'st  and 
usual  mode  of  rupture  can  be  developed  only  by  over- 
turning the  pier  or  lower  part  of  the  arch.  While  the 
first  mode  of  rupture  can  be  prevented  by  giving  to  the 


..82 


THEORY    OF    THE    ARCH. 


pier  a  sufficient  mass,  or  by  opposing  to  the  horizontal 
thrust  the  siniihir  thrust  of  ani^ther  arch,  the  tliirtl  can  be 
rendered  impossible  only  by  changing  the  proportions  of 
the  arch  itself 

Another  interesting  fact  is  to  be  noticed.  The  third 
mode  of  rui)ture  gives  rise  to  two  new  joints  of  rupture, 
each  about  lialf  A\'ay  between  the  key  and  the  lower 
joints. 


Fig.  43. 


Let  it=:the  radius  of  the  extrados;  ?'=the  radius  of 

the  intrados;  K=^  —  ;  v=^i\\Q  angle  m  CI  corresponding 

to  the  maximum  F;  v'—t\iQ  angle  iii  Ch  corresponding 
to  the  minimum  F' . 

The  following  table  gives  the  values  of  F^  F\  x\  v' 
corresponding  to  very  light  arches,  or  small  values  of  JC 


Values  of 
r 

Ratio  of  tbe 
diameter 

to  the 
tliiekncss. 

Values  of  ».  and  F. 

i 

Values  of  f' nnJ  F'. 

r. 

F. 

i 

F\ 

1.00 
1.01 
1.02 
1.03 
1.04 

200.0 

100.0 

60.^ 

50.0 

72° 

71 

70 

68 

67 

r»x  0.05561 
"   0.00214 
"    0.00808 
"    0.07526 
"    0.08188 

23° 

28 

35 

r'x  0.02177* 
"   0.04349 
"   0.06452 
"   0.08486 

*  Ijy  tlic  law  of  Jifferencei 


i 


LIMIT    OF    POSSIBLE    ARCHES.  383 

The  lowest  two  numbers  in  tlie  last  column  have  been 
taken  from  the  calculations  of  M.  Petit. 

Corresponding  to  J{^=1.02^  we  see  that  i^is  more  than 
half  as  large  again  as  jF"  ;  in  other  words,  an  arch  of  such 
proportions  is  far  below  the  limits  of  possibility.  When 
X=  1.03,  the  excess  oi  F  oyqv  F\  is  much  less.  When 
X=1.04,  F  is  less  than  F'.  By  proportional  parts,  we 
find  these  two  forces  to  be  equal  when  jff'^  1.0378,  or 
when  the  thickness  is  -^^  of  the  span.  In  this  case,  v=-^l° 
and  'y'=33;i-°,  nearly. 

Any  lighter  arch  surcharged  horizontally,  will  fall  by 
the  third  mode  of  rupture  whatever  be  the  thickness  of 
the  piers. 

M.  Petit  has  given  an  erroneous  solution  of  this  ques- 
tion, by  supposing  the  horizontal  thrust,  acting  at  the 
intrados  of  the  key  at  the  moment  of  rupture,  to  be  gener- 
ated by  the  tendency  of  the  whole  semi-arch  to  rotate 
forward  towards  the  center.  In  this  way  he  has  greatly 
under-estimated  the  thrust,  and  given  a  limit  of  thickness 
nearly  as  small  as  -g\  the  span.  Other  French  writers 
Jiave  fallen  into  the  same  error. 

But  M.  Petit  made  a  greater  mistake  in  supposing  the 
question  to  be  more  curious  than  useful.  It  is  the  begin- 
ning, as  we  shall  see,  of  a  highly  important  investigation. 

No  branch  of  the  subject  can  be  more  important  than 
that  which  determines,  and  shows  how  to  determine,  cor- 
rectly, the  limits  of  practicability.  In  this  country  and  in 
Great  Britain,  these  limits  are  frequently  approached— 
sometimes  passed.  In  France,  arches  seem  to  be,  in 
general,  of  heavier  proportions. 


LIMIT   THICKNESS    OF   POSSIBLE   SEGMENTAL    ARCHES. 

143.  If  the  arch  be  segmental,  and  the  semi-angle  at 
the  center  less  than  the  angle  of  rupture,  the  limit  will 


384  THEORY     OF    THE    ARCU. 

become  smaller,  diminishmg  more  and  more  as  tlie  open- 
iui;  (limiiiislies. 

The  following  table  gives  tlie  limit  thickness  of  seg- 
mental arches  surcharged  horizontally,  np  to  the  level  of 
the  extrados  at  the  key,  on  the  supposition  that  mere  edges 
of  masonry  can  withstand  any  pressure  whatever. 

F  and  V  are  the  same  as  in  the  preceding  table.  F 
is  the  ultimate  thrust  in  the  third  mode  of  ru])ture,  the 
point  of  application  being  at  the  intrados  on  the  key.  It 
is  taken  from  table  E'.  Let  ?'=ithe  radius  of  the  intrados ; 
5'=the  span;  /=the  rise;  (7=the  coefficient  of  i^  in  table 
E'.     Then 

F^i^y.C{\-\-'-{K-V)), 

J 

The  values  of  K  are  written  at  top. 

In  each  horizontal  division,  will  be  found  the  limit 
value — of  J5r,  of  the  diameter  2r  in  terms  of  the  thickness 
d—r  (^—1),  and  of  the  span  s  also  in  terms  of  d. 


LIMIT    OF    POSSIBLE    ARCHES. 


LIMIT  THICKNESS  OF  POSSIBLE  SEGMENTAL  ARCHES,  ON  THE 
SUPPOSITION  THAT  MEKE  EDGES  OF  MASONRY  CAN  WITH- 
STAND   ANY   PRESSURE   WHATEVER. 


I 


Values  of  K 

1.01 

1.02 

1.03 

1.04 

Values  oiv' 
Values  of  F 

18"      * 
r^x  0.02177 

23° 
r^'x  0.04349 

28° 
r^'x  0.06452 

35° 
r'xn.(j8486 

s=4/;-=2.50;  Fz= 

{  7r=  1.0342 
Limit  •{  2r=58ici 
(    s=47x(Z 

r^x  0.06200 

r^'x  0.06982 

r^x  0.07758 

.5  =  5/;  -=2  9  !       F= 

r  ^=1.0275 
Limit  "I  2?"=:73  xd 
{    s=50J 

r=x  0.05288 

r'^x  0.06138 

r''x0.06982 

«=6/;j=5;         F= 

(  A'=  1.02207 
Limit -j  -lr=m^d 
{    s—bU 

r^x  0.03707 

r'^x  0.04600 

r^x  0.05491 

.=7/;^=Y;     F= 

(A'=  1.0177 
Limit  •]  -Ir^UU 
(    s=60(i 

r'x  0.03137 

r'^x  0.04058 

r^x  0.04974 

,s=8/;^=8.50;  F= 

(  7i'=  1.0144 
Limit-]  2r=139j 
(    s=&od 

r^x  0.02718 

r'x  0.03654 

1 

s=10/;  -=13;  i^= 

(A'=  1.0099 
Limit ■{  2>— 202(Z 
(    s=78rf 

r^'x  0.02164 

r'^x  0.03124 

-^%^z: ' 

*  v'  and  F'  for  ^=1 .01  have  been  determined  by  the  law  of  diffei'onces. 


3SG 


TIIEOUY    OF    THE    ARCH. 


LIMIT   THICKNESS    OF   PRACTICABLE   ARCHES. 

144.  In  the  preceding  articles  142  and  143,  we  liave 
treated  of  tlie  ultimate  thrust.  At  the  moment  of  rupture 
mere  edges  or  points  of  masonry  were  supposed  to  be  in 
contact,  and  able  to  withstand  all  the  pressures  that  might 
be  thrown  upon  them.  We  have  given  too  high  a  limit, 
for  it  is  evident  that  these  edges  will  give  way  long  before 
the  pressures  come  upon  mathematical  points. 

Let  us  now  determine  the  least  ^yrrtc'//c'(//>/^  thickness  of 
the  semicircular  arch,  surcharged  horizontally  up  to  the 
summit  of  the  extrados,  on  the  condition  that  the  curve  of 
pressure  shall  be  everywhere  traced  between  two  other 
curves  which  divide  the  joints 
into  three  equal  parts.  The 
case  is  analosfous  to  the  one 
already  considered,  art.  142, 
the  extrados  and  intrados  of  the 
latter  case  being  now  replaced 
by  the  limit  curves. 

The  curve  of  pressure  touch- 
ing the  inferior  curve  at  <?,  on 
the  central  joint,  and  again  at  ^, 
on  the  joint  of  maximum  thrust, 
must  not,  at  any  intermediate  point,  t\  pass  outside  the 
superior  limit. 

The  general  expression  of  the 
horizontal  force  F  which,  applied  at 
any  point,  i?',  on  the  central  joint, 
shall  keep  the  segment  h  a  m  n  r  h  in 
equilibrium  on  any  point,  /,  of  any 
joint,  m  //,  is 


Fio.  44. 


F=\r 


sin.  w 


R'  ir 

cos.w 

r      r 


i  ^U--zK-V^--2K)m^.{I■\-v))- 
{  sin./^  r  ^  r  ' 


Sv 

sin.y 


X-  + 


cos 


s.'i"  f 


(68) 


LIMIT    OF    PRACTICABLE     ARCHES.  387 

in  wliicli  E'=CE';  r'=Qr' ;  v^angle  r'  C FJ  or  m  Oh; 
J=tlie  angle  between  the  roof  and  a  vertical ;  /'^tlie 
radius  of  the  intrados ;  Kr—\}viQ  radius  of  the  extrados. 

Suppose  7=90°;  B! ^r' =r-\-\i\K-X)^\r{:2^K\ 
Substituting  these  values,  (68)  is  reduced  to 

F-r^  \  ^-^ ^/-— — ^ ivcot.lv  c        (09) 


The  maximum  value  of  this  expression  evidently  gives 
the  actual  thrust  for  the  case  in  hand. 
Suppose  jK'=^r(2H-^),  as  above, 

and  /  =  r+|;</i--l)  =  K2/i^+l); 
(68)  reduces  to 

,Ad\\h'{\  —  \cof,.v)K'' -^^iimhK^ —\v m\.v —V s,m.vK+{l  —Q.OS.V)  \ 
~*"  1  (2  — COS.V)  — A"(2  COS.V  — 1)  \ 

(70) 

The  minimum  value  of  this  expression  is  the  least  force 
which,  acting  horizontally  at  c^hc-=^^ha^  shall  cause  the 
resultant  of  the  force  and  of  the  weight  of  the  segment 
h  a  m  n'  r  h^  which  corresponds  to  the  minimum,  to  reach 
the  superior  limit.  Any  greater  force  would  carry  this 
resultant,  and  consequently  the  curve  of  pressure,  beyond 
the  superior  limit. 

It  is  obvious  that  the  greatest  value  of  i^in  any  arch 
should  not  exceed  the  least  value  of  F'.  If  these  two 
forces  be  equal,  the  curve  of  pressure  will  touch  the  supe- 
rior limit,  as  represented  in  fig.  44. 

Let  ii^=:the  maximum  value  of  F  in  (69) ;  ?'=:the 
angle,  mCh^  corresponding  to  that  maximum;  i<"=the 
minimum  value  of  F'  in  (70)  ;  -?;'=::  the  corresponding 
angle. 

The  following  table  gives  the  values  F^  v,  F\  and  v\ 
as  far  as  necessary  for  the  object  in  view : — 


588 


THEORY    OF    THE    ARCH. 


Yalae  of 

r 

Ratio  of  the 
diameter 

to  the 
thickness. 

Va 

ucs  of  F  and  «. 

Values  of/*' and  p*. 

r. 

f-                 1 

'C. 

F'. 

1.01  - 

200.00 

1.02 

100.00 

70° 

/•«x  0.07012 

il5° 

r'x  0.03175 

1.03 

66i 

68 

"     0.07754 

120° 

"     0.04758 

1.04 

50.00 

67 

"     0.08510 

22°  30' 

"     0.06317 

1.05 

40.00 

65 

"     0.09275 

25° 

"     0.07873 

1.06 

331 

64 

"     0.10055 

30° 

"     0.09420 

1.07 

28.57 

62 

"     0.10848 

32°  30' 

"     0.10930 

1.08 

25.00 

\Ve  learn  from  this  table  that  F  is  greater  than  F\ 
that  i:«,  that  the  arch  is  impractieahle,  for  all  values  of  K 
less  than  1.06 ;  and  that  F  is  less  than  F\  or  the  arch 
practicable,  for  /f=1.07  and  all  larger  values.  By  pro- 
portional parts  we  find  these  two  forces  to  be  equal  Avhen 
/i^=:1.0G880,or  when  the  span  is  about  29  times  the  thick- 
ness. AVe  thus  find  the  least  possil)le  arch  which  can 
exist  without  any  opening  of  the  joints. 


145.  If  the  arch  be  segmental,  and  the  semi-angle  at 
the  center  less  than  the  angle  of  maximum  thrust,  the 
limit  of  practicable  arches  will  of  couree  become  smaller, 
decreasing  more  and  more  as  the  opening  diminishes. 

The  following  table  gives  the  limit  thickness  of  seg- 
mental arches  surcharged  horizontally  up  to  the  level  of 
the  extrados  at  the  key.  F'  and  v  are  the  same  as  in  the 
preceding  table,  i^is  the  actual  thrust,  on  the  supposition 
of  a  curve  of  pressure  touching  the  inferior  limit  at  the 
key  and  at  the  springing  line.  F  is  obtained  from  table 
EE  by  tlie  following  formula. 
in  that  table, /=  the  rise,  then 


Let  6— the  coefficient  of  i^ 


F^i'xC 


LIMIT    OF    PRACTICABLE    ARCHES. 


389 


LIMIT  THICKNESS  OF  SEGMENTAL  ARCHES,  ON    THE    CONDITION 
THAT   NO   JOINT   SHALL   BEGIN   TO    OPEN. 


Values  of  K. 

1.01 

1.02 

1.03 

1.04 

1.05 

i.oe' 

1.07 

Values  of  v'=z 
Values  of  i^'=r-x 

12°   * 
). 01568 

15° 

0.03175 

20° 
9.04758 

22°  30' 
). 06317 

25° 
0.07873 

30° 
0.09420 

32°  30' 
). 10930 

5=4/;  y= 2. 50;  F=r^x 

(  A'=1.0G521 
Limit  •{  2r=31xrf 
(    a—-l5xd 

).09750 

).  10623 

r      29 
s:=5f;j=~;F^r^x 

(Z'=  1.0545 
Limit -{  2r=37xfZ 
(    s=25xd 

0.07240 

0.08157 

0.09072 

s=e>f;-  =  5;  Fz=r''x 

(  A'=  1.0441 

Limit  -|  2c=45  x  d 

(    5=27x0? 

0.05625 

). 06564 

0.07520 

0.08466 

r     53 
s^1f;y^-^;F^r^x 

^A'=  1.03544 
Limit  ■<  2r=56  xd 
(    s—Wxd 

0.04123 

0.05083 

0.06045 

.s=Sf;j-8.50;F=zr''x 

(A'=  1.0288 

Limit  -1  2r=:70  x  d 

(    s=33xd 

0.03707 

0.04684 

s=:10/;  j=U;F=r^x 

i  A'=l. 01976 
Limit-]  2r=:101xc? 
(    s=39xc? 

0.02 ISC 

0.03161 

.s=lt5/;y=32.5;i^=r^x 

(A'=  1.0088 
Limit-^2r=227xJ 
(    s=56xd    . 

0.01496 

0.0250E 

*  v'  and  F'  for  A'=l  .01  determined  by  tlie  law  of  differences. 


390  THEORY    OF    THE    ARCH. 

EFFECT    OF   SURCHARGE   TPON   THE   PRACTICABILITY   OF 

ARCHES. 

14(').  Returning  to  tlie  seraicircnlar  arch  surcliarged 
as  above,  let  ns  illustrate  the  effect  of  a  surcharge  of  a 
constant  depth  above  the  extrados  at  the  key. 

When  ^=1.04,  or  the  span =50  times  the  thickness, 
we  learn  from  art.  144,  that  the  arch  is  impracticable  ;  but 
adding  a  surcharge =y'y  the  radius,  or  -^^  the  span,  we  find 
the  arch  to  become  barely  practicable,  the  lower  joint  of 
rupture  Ijeing  58°  from  the  key,  and  the  upper  joint  30°. 

The  addition  of  a  deeper  surcharge  up  to  a  certain 
point,  would  increase  the  stability  of  the  arch  ;  that  is, 
cause  the  curve  of  pressure  to  dej^art  less  from  the  central 
line  of  the  joints. 

The  addition  to  the  actual  thrust  caused  by  a  surcharge 
of  constant  depth,  t,  is,  for  any  angle  t), 

to  be  added  to  the  value  of  ^,  eq.  (69),  corresponding  to 
the  same  value  of  v.  The  maximum  value  of  I^^-\-A  to  be 
obtained,  as  usual,  by  variations  in  v. 

In  (71)  as  well  as  in  (69),  the  point  of  application  of 
the  horizontal  thrust  at  the  key  is  supposed  to  be  at  ^  the 
length  of  the  joint  from  the  intrados,  Avhile  the  curve  of 
pressure  is  supposed  to  be  at  the  same  distance  from  the 
intrados  at  the  joint  of  maximum  thrust. 

The  effect  of  surcharge  to  be  added  to  i^',  eq.  (70),  is, 

the  minimum  of  jF"-\-A'  to  l)e  obtained  byvai'iations  in  v. 
In  (70)  and  (72)  the  point  of  api)lication  upon  the  key 
is,  as  above,  at  -i-  the  length  of  the  joint  from  the  intrados, 
and  at  the  joint  of  minimum  thrust  at  the  same  distance 
from  the  extrados. 


CURVE    OF    PRESSURE. 


391 


EQUATION  OF  THE  CURVE  OF  PRESSURE  IN  THE  ARCH. 

147.  Assume  as  known  tlie  lior- 
izontal  thnist,  F^  and  it',  its  point 
of  application. 

"  ^''^C'r'^the  variable  distance 
of  tlie  curve  of  pressure  from 
the  center  C. 

"    'y=:tlie  angle  t  C  E'. 

"  ^;r=tlie  horizontal  distance,  g'  f,  rro.  46. 

between  the  vertical,  Ca,  and  the  center  of  gravity 
of  the  segment  a  m  n  r  s  a. 

"  /S'^^the  surface  atiinrsa^  bounded  by  the  circular 
arc  a  m.,  the  verticals  G IH,  s^  n  r^  the  joint  7ii  ti,  of 
any  length,  and  the  upper  surface  s  7\  of  any  incli- 
nation. 

As  the  thrust,  F^  and  the  weight,  8^  are  in  equilil3rium 
on  every  point  of  the  curve  of  pressure,  we  have  the 
equation  of  moments, — 

F{It'—r' COS.  v)—SQ'  sin.  v— />), 
giving 

FxB'+Sxp 


'/Sx  sin.  v-\-Fx  COS.  v 


(73) 


F  must  l3e  found  by  calculation,  or  by  the  taldes  con- 
tained in  this  paper ;  it  is  the  actual  thrust. 

i?',  the  point  of  application,  and  the  distance  CIi',=Ii\ 
must  be  assumed  according  to  the  circumstances  of  the 
case.  S2:>  is  the  moment  on  C  of  the  surface  0  n  rsC 
minus  the  moment  on  G  of  the  sector  m  Gcl  8  is  the  sur- 
face Gnr  s  (7— the  sector  m  Ga. 

Assuming  any  particular  value  of  t',  8})  and  8  are 
easily  calculated.  A  table  giving  the  values  of  the  surface 
Gma—\vr'^^  and  its  moment  on  <7,=^^'Xl— cos. t'),  at 
14 


392  THEORY    OF    THE    ARCH. 

intervals  of  .">  or  10  degrees,  would  greatly  fiu-illtate  tlie 
calculation  of  r'. 

Su})pose  we  are  in  doul)t  whether  the  proposed  arch 
is  practicable  or  not.  This  doubt  need  not  ai-ise  unless 
the  pi'oposed  arch  is  very  light,  and  surcharged  horizon- 
tally, or  nearly  so. 

Determine  the  thrust  on  the  condition  that  the  curve 
of  pressure,  at  the  key  and  the  lower  joint  of  rupture,  shall 
touch  the  inferior  limit  one-tliird  the  length  of  the  joint 
from  the  intrados.  Still  better,  ascertain  the  somewhat 
larger  thrust  which  is  given  by  table  DD,  or  the  formulae 
of  art.  11 G,  on  the  supposition  of  a  curve  of  pressure  pass- 
ing through  the  middle  of  the  joints  in  question. 

Assume  M' ^=zr-\-\(^Il—i-)^  and  com^mte  several  values 
of  ;*'  near  the  bisectuig  line  of  the  angle  i?'  C  r\  mn  being 
the  joint  of  greatest  thrust. 

Should  the  greatest  value  of  r  exceed  7'-|-|(7t— r),  we 
know  that  the  proposed  arch  is  impracticable. 

Should  the  greatest  value  of  r'  be  less  than  r+-|(it  — r) 
and  larger  than  r-{-^(^R—r\  we  know  that  the  arch  is 
practicable,  and  ai'e  at  liberty  to  suppose  a  more  elevated 
curve  of  pressui'e,  so  traced  as  nearly  to  equalize  the  pres- 
sure inside  and  outside  of  the  central  line  of  the  joints. 


POINT    OF    APPLICATION    OF   TTFE   TIIRLST   AT   THE   KEY. 

148.  The  formula3  used  in  the  investigation  of  circular 
arches  aie  transcendental  in  foi'm, so  that,  in  general,  prac- 
tical conclusions  can  be  di'awn  only  from  laljorlous  calcula- 
tions. 

We  are  about  to  meet  with  an  exce])tioii,  and  shall 
reach  at  once  some  generalizations  of  great  value  and 
interest  with  little  labor. 

We  liave  assumed  that  the  ]K)int  of  a])plicati(»n  of  the 
thrust  at  the  key  shall  be  somewhere   in   the  middle  s])ace 


POINT    OF    APPLICxVTION. 


393 


formed  by  dividing  tlie  vertical  joint  into  three  equal 
parts.  We  have  found  in  certain  cases,  that  the  curve  of 
pressure  must  start  from  the  lower  limit  to  keep  inside,  the 
upper-limit  curve. 

This  takes  i)lace  only  in  very  liglit  arches  loaded  liori- 
zontally  or  nearly  so.  As  we  increase  the  arch  or  increase 
the  surcharge,  we  are  evidently  at  liberty  and  in  fact  com- 
pelled to  suppose  the  curve  to  start  from  a  higher  point. 

Let  us  now  inquire  w^hen  we  are  at  liberty  to  suppose 
this  point  of  departure,  or  point  of  application  of  the  thrust 
at  the  key,  to  be  as  high  as  the  upper  limit. 

Table  DDD  gives  the  thrust  corresponding  to  this  sup- 
]30sition  for  the  arch  surcharged  horizontally,  the  curve  of 
pressure  touching  the  inferior  limit  at  the  joint  of  great- 
est thi'ust.  It  is  obviously  necessary 
that  the  curve  of  pressure,  starting 
from  the  upper  limit  c?,  ca—^ha^ 
should  not  at  any  point  a1)ove  the 
joint  of  greatest  thrust,  m  n^  be  found 
outside  of  the  circular  arc  of  w^hich 
Co  is  the  radius.  From  the  point  c 
it  should  run  immediately  below,  and 
not  above,  that  arc. 

In   ((58)   suppose  i?'=:r'=:r+|(^-^')  =  iKl+2^0  ; 
7=zO,  and  -y^O.     We  have 

(i.)=,.^^!+Z!_^'.  (74) 

Under  the  same  suppositions  we  find  the  effect  of  sur- 
charge to  be 


(75) 

1+2^  ^     ^ 

The  1st  column  of  the  following  table  contains  the 
values  of  K;  the  2d  and  3d  columns  the  values  of  {F) 
and  {A)  respectively;  the  4th  and  5th,  the  horizontal 
thrust  and  the  effect  of  surcharge,  both  taken  from  table 
DDD. 


394 


THEORY     OF    THE     ARCH. 


ao 

(F)  from  0( 

1.  (T4>  and 

1 
1 

Valuos  ..f  F 
Tuble 

and  A  from 
DUD. 

! 

\<- 

(/-) 

(-1) 

F 

A 

1 

1.02 

/•-  X 

0.02026 

r<x  1.01.33 

r^  X 

0.0G94 

r^xO.8971 

1.04 

0.0410G 

"    1.02G5 

ii 

0.0832 

"    0.8612 

1 

LOG 

0.00238 

"    1.039G 

!       u 

0.0973 

"    0.8358 

1.08 

0.08421 

"    1.0527 

1       '' 

0.1112 

"    0.8156 

1 

1.10 

0.10G56 

"    1.0G5G 

1       " 

0.1250 

"    0.7988 

1.1-2 

0.12942 

"    1.0785 

1       '^ 

0.1387 

"    0.7841 

! 

1.14 

0.15278 

'•    1.0913 

!       (( 

0.1523 

"    0.7712    i 

1.00 

0.22588 

"    1.1294 

i 

0.1918 

"    0.7387 

We  learn  from  this  table,  that  when  K  is  small  F  is 
much  larger  than  (i^).  In  snch  cases,  if  the  curve  of 
pressure  were  to  start  at  r,  it  would  immediately  run 
above  the  upper  limit.  When  /i^=1.14,  F  is  nearly 
equal  to  (i^)  ;  for  greater  values  of  K^  F'ls  less  than  {F). 
Consequently,  for  all  values  of  ^exceeding  1.14,  we  are 
at  liberty  to  suppose  a  curve  of  pressure  starting  from  the 
upper  limit  on  the  vertical  joint. 

We  have  found  that  when  F  is  less  than  1.06886,  the 
arch  is  impracticable,  art.  136  ;  that  when  F  is  equal  to 
1.06886,  the  arch  is  barely  practicable,  the  curve  of  pres- 
sure necessarily  starting  from  tlie  lower  limit  on  tlie  key. 

Between  7r=  1.06886,  and  /ir=:1.14,  the  starting  point 
will  gradually  rise  from  the  lower  to  the  upper  limit. 

By  comparing  the  values  of  (7*^)  given  above,  with 
those  of  F  in  table  EE,  segmental  arclies,  avc  find  that 
the  curve  of  pressure  may  start  from  the  u}>])('r  limit  on 
the  key,  as  follows : 

6-=  r/or  f=43°  36'  10",  when  7l  =  1.12,  nearly. 


s=  C/or  t'=36°  52' 10", 
6-=  If  orv=Sl°  53' 27", 
6-=  8/ or  7— 28°  4' 20", 
.9=lCjf  or  v^W  15' 00", 
and  for  all  larger  values  of  F. 


F=1.\0, 
F=l.OS, 
F=U)7, 
F=l.02, 


rOINT    OF    APPLICATION. 


However  small  the  value  of  K^  hy  giving  to  t  in  (A), 
and  A^  a  certain  value,  we  shall  make  {^F)-\-(^A)-=F-\-A. 

Example.     When   A^=1.02,  suppose   2;=rX0.42,   we 
have  (i^)+(^4)  =  >-^X.44r)8, 
and    i^+^4=7''X.4461. 

When  /i  =  L04  and  i^=rX0.25,  we  find  the  two  sums 
to  lie  nearly  equal. 

These  results  agree  with  those  already  obtained  by 
independent  means,  and  show  that  an  arch  impossible  or 
imi)racticable  without  surcharge,  may  become  perfectly 
safe  when  a  load  of  sufficient  depth  has  been  added. 

We  will  add  a  word  of  explanation  as  to  the  principle 
of  the  demonstration  above  given. 

The  quantity  which  reduces  to  {F),  equation  (74), 
when  v=o^  is  the  variable  horizontal  force  which,  applied 
at  <?,  ac=^%ah,  shall  hold  any  segment  a  mn  t  h  a  in  equili- 
brium on  r,  fig.  47,  Cr=Cc^r-^%{E-r).  Let  (i^)  = 
that  variable  force. 

In  like  manner,  the  quantity  which  reduces  to  (^4), 
equation  (75),  when  v=o^  is  the  variable  effect  of  sur- 
charge under  the  same  supposition  as  to  the  points  of  appli- 
cation.    Let  (J.)'=that  variable  effect. 

The  sum  (i^)  +  (^),  corresponding  to  v=o,  will  either 
be  a  maximum  or  a  minimum  ;  generally  the  latter. 

In  either  case,  the  sum  corresponding  to  small  values 
of  V,  will  not  differ  sensibly  from  {F)  +  {A),  to  which  it 
is  reduced  when  ^1  =  0. 

Consequently,  if  the  actual  thrust,  (i^+^4),  exceed 
([F)j^[A)),  it  will  also  exceed  the  neighboring  sums  of 
(^(^F)' -^{A)'),  and,  so  long  as  this  superiority  continues,  the 
curve  of  pressure  must  run  entirely  outside  the  superior 

On  the  other  hand,  if  {F+A)  be  less  than  {{F)^-{A)) 
the  curve  of  pressure  will  remain  within  the  superior  limit 
at  all  points  above  the  joint  of  maximum  thrust. 


396  THEORY     OF    THE    ARCH. 

POIN'T    OB'   APPLICATION    OF   THE   ULTIMATE    THRUST. 

140.  Bv  a  course  of  inve.-^ti^Ci^tion  entirely  similar  to 
that  indicated  above,  we  may  prove  that  the  point  of 
application  of  the  ultimate  thrust  can  not  always  1)e  at  the 
extrados,  but  must,  in  many  arches,  be  below  it.  When 
/r=  1.03 78,  this  point  is  necessarily  at  the  intrados,  art. 
142,  and  K  must  be  increased  to  a  certain  extent  before 
it  can  rise  to  the  extrados. 

Some  of  the  tables  of  the  ultimate  and  of  the  actual 
thrust  are  based,  in  reference  to  light  arches,  u]wn  an  im- 
possible supposition  as  to  this  point  of  apjjlication ;  but 
their  value  is  not,  on  that  account,  seriously  impaired,  be- 
cause the  thrust  is  not  sensibly  affected  by  small  changes 
in  the  j^oint  of  aj^plication. 


EE5IAEKS    ON    TABLE   L 

150.  This  table  gives  tlie  ])i'inei})al  elements  of  many 
celebrated  bridges :  the  dimensions,  the  total  thrust  at  the 
key  in  terms  of  the  radius  of  the  intrados,  the  mean  pres- 
sure, in  pounds,  on  each  square  foot  of  the  key,  and  the 
i-atio  of  the  estimated  ultimate  resisting  power  of  the 
material,  to  the  pressure  on  the  edge  most  exposed.  This 
j^ressure  is  assumed  to  be  double  the  mean  pressure. 

Tlie  thrusts  have  all  been  obtained  from  the  tables  and 
formulfB  contained  in  this  paper, — circular  arches,  from 
table  DD,  No.  3  excepted,  Avhich  is  taken  from  table 
DDD ;  elliptical  arches  from  table  I)D,  as  exi)lained  in 
art.  121 ;  segmental  arches,  from  art.  IIG. 

In  the  absence  of  definite  information  as  to  the  load 
borne  by  the  several  arches,  it  was  necessary  to  make 
some  hypothesis.  Very  few  of  them  were  loaded  with 
masonry  over  tlie  reins  up  to  tlie  level  of  the  top  of  the 
key  ;  but  all  had  some  surcharge  over  the  key.     It  has 


REMARKS    OF    TABLE    I.  397 

been  assumed  in  all.  cases,  that  the  surcliarge  was  ec^uival- 
ent  to  a  load  in  masonry,  rising  throngliont  to  tlie  level  (»f 
of  the  top  of  the  key-stone. 

The  results  are  given  as  illustrations,  and  as  approxima- 
tidns  to  the  ti-uth.  The  reader  is  cautioned  against 
receiving  them  as  fixed  or  established  facts. 

No.  1.  Bishop  Auckland  Bridge.  r=G8'.29 ;  K— 
1.027;  .5-=54.77x^/;  2r=74.50Xf/;  .y=4.57x/.  When 
^=4/"  we  have,  as  the  limit  of  possibility,  art.  143,  K— 
1.0342;  when  s—^f  we  have,  at  the  limit,  7^=1.0275; 
hence,  by  proportional  jiarts,  the  limit  corresponding  to- 
5-=4.57,  is  found  to  be,  /f=  1.0304.  Therefore,  a  bridge 
of  the  given  dimensions,  and  of  the  supposed  load,  would 
fall,  by  the  third  mode  of  rupture,  whatever  the  stability 
of  the  piers,  and  whatever  the  resisting  power  of  the 
materials. 

The  date,  1388,  renders  it  probable  that  this  bridge 
was  but  lightly  loaded  at  the  reins. 

No.  2.  Llanwast  Bridge.  7^=33'.24  ;  7^=1.045;  .v= 
38§7;  2r=44.30x7;  6  —  3.41/.  The  limit  values  oi K 
are  nearly  the  same  as  in  semicircular  arches ;  viz.,  the 
limit  of  possibility,  art.  142,  7^=1.0378;  limit  of  practic- 
ability, art.  144,  ^=:  1.06886.  This  bridge,  therefore, 
under  the  suj^posed  load,  is  impracticable,  though  not 
impossiljle. 

It  could  not  exist  without  consideralde  cracks  at  the 
the  key,  the  springing  line,  and  the  intermediate  joints  of 
rupture.  We  have  doubtless  overestimated  the  load  upon 
the  reins. 

No.  3.  Westminster  Bridge.  ^'=38';  7^=1.20;  2;'= 
10c/.  This  is  a  very  heavy  structure,  of  three  times  tlie 
required  thickness,  whether  we  look  at  the  limit,  art.  144, 
or  at  the  pressure  per  unit  of  surface  at  the  key,  table  I. 


398  THEORY    OF    THE    ARCH. 

No.  4.  Taaf  Bridge.  r=SrM ;  A^r=1.029  ;  .5  =  5G  X^/; 
2/'=70X</,•  ^'=4:/.  Limit  of  possibility,  art.  143,  /r= 
1.0342.  This  arch,  therefore,  under  the  supposed  load,  is 
impossible.  In  fact,  this  l>ridge  fell  on  the  removal  of  the 
center,  but  was  rebuilt  witli  a  diminished  load  n})i)n  the 
reins. 

No.  5.  Wellington  Bridge,  over  the  Aire.  r=90'.83  ; 
A^=1.033;  6'=?.3JV;  2;'=G0.r)5X'^-  -s-^f.  Limit  of 
l^ossibility,  see  table  in  art.  143,  7^=1.019  ;  limit  of  prac- 
ticability, see  table  in  art.  145,  /i"=  1.0383.  Not  quite 
practicable  under  the  supposed  load,  if  we  suppose  the 
thickness  of  the  arch  to  be  the  same  throughout,  but  more 
than  practicable  when  we  take  into  consideration  the 
rajjidly  increasing  thickness  on  each  side  of  the  key.  An 
increase  of  ^  instead  of  IJ  at  the  springing  line,  would 
have  made  this  arch  more  tlian  practicable  and  safe  under 
the  worst  possible  load. 

No.  G.  Waterloo  Bridge.  r=GO';  /=35' ;  .7=4'.T5; 
the  thrust,  art.  121,  is  nearly  equal  to  that  of  a  circular 

arch  of  the  same  span,  and  of  a  thickness  =  ,7'/=  8'.  143. 

The  curve  of  pressure  has  nearly  the  same  relative  situation 
in  both,  and  they  are  alike  exposed  to  the  third  mode  of 
rupture.  Li  the  circnlar  arch  we  liave  A^=1.1357,  which 
places  both  arches  far  al)ove  the  limits  of  practical)ility. 

It  is  worthy  of  remark,  tliat  a  segmental  arcli  of  the 
same  rise  and  s])an,  and  of  tlie  same  thickness,  equal 
throughout,  would  be  barely  ])racticable.  In  tliis  areli  we 
sliould  have  /— G8'.93,  and  A.^=:1.0G89. 

The  intrados  of  the  segmental  ai'cli  departs  more,  in  its 
general  direction,  from  the  ciu-ve  of  pressure,  than  tlie  intra- 
(h)S  of  the  ellipse. 

No.  7.  Louden  l>iidge.  This  bold  and  beautiful  woi-k 
lias  the  thrust  and  the  geneial  character  of  stability  within 


REMARKS    ON     TABLE    I.  399 

itself,  of  a  circular  arch  of  tlie  same  span,  and  <>f  such 
thickness  as  to  make  the  ratio  J^of  the  two  radii  e(|\uil  to 
1.1316,  or  the  span  aljout  fifteen  times  the  thickness. 

A  segmental  arch  of  the  same  rise,  span,  and  thickness, 
equal  throughout,  ?'=95',  7^=:  1.05263,  would  not  be  prac- 
ticable, but  might  l)e  made  so  by  increasing  the  thickness 
gradually  on  each  side  of  the  key,  to  T'.SO  at  the  spring. 

No.  8.  Staines  Bridge.  Span =8  times  the  rise  ;  K= 
1.038.  More  than  practicable  and  secure  without  any 
increase  of  thickness  between  the  key  and  the  spring  (see 
art.  115). 

No.  9.  Chester  Bridge.  r=140'.  nearly;  Jf=1.0286  ; 
mean  value  of /!■=  1.0357,  nearly;  5=:50Xt/;  2?'=70X^/; 
.5-=: 4.76/.  Impracticable  and  but  little  more  than  possible, 
if  loaded  according  to  our  supposition.  AVe  are  told  that 
the  crown  settled  only  2^  inches  on  the  removal  of  the 
center.  This  is  no  proof  of  excellence  of  w^orkmanship. 
This  arch  is  near  the  third  and  not  the  first  mode  of  rup- 
ture. Tlie  crown  tends  to  rise,  or  has  little  tendency  to 
settle  down,  on  the  removal  of  the  center. 

No.  11.  Hutchesou  Bridge.  6-=5.925x/;  /t'=1.0537. 
Perfectly  secure  without  any  increase  of  thickness  between 
the  key  and  the  spring. 

No.  12.  Whitadder  Bridge.  6-=6.52/;  /i'=1.0374; 
mean  value  of  7^=1. 0411.  Nearly  practicable  without 
any  increase  of  thickness  between  the  key  and  the  spring. 

No.  13.  Kail  way  Bridge  at  Maidenhead.  This  remark- 
able structure  is  made  of  bricks.  It  has  the  span  and 
thickness  at  the  key,  of  the  celebrated  bridge  of  Neuilly, 
and  a  rise  considerably  less.  It  is,  perhaps,  the  boldest 
work  of  which  we  have  any  record.     It  has  the  thrust  and 


400  THEORY     OF    THE    ARCH. 

the  general  stability  of  form  of  a  circular  arcli  of  the  same 
span,  and  of  a  thickness  =  13'.86  thronghont,  corresponding 
to /r=r  1.2165. 

The  ultimate  resisting  ])ower  of  the  material  is  only  3i 
times  the  actual  pressure  at  the  most  exposed  ed^es. 

Xo,  14.  Bridge  of  Xeuilly.  Very  similar  in  its  pro- 
portions, to  the  London  Bridge,  but  a  little  l)older  when 
vre  take  into  consideration  the  strensT^th  of  the  material 
(see  the  last  column). 

It  has  nearly  the  thrust  and  stability  of  a  circular  arch 
of  the  same  span,  and  of  the  constant  thickness  of  10 i  feet, 
corresponding  to  7r=1.164. 

Xo.15.  The  Bridge  of  Pesmes.  7'=67'.11;  6  — 11.G7X 
/;  /r=:1.0.")7.  A  very  heavy  work.  One  half  the  thick- 
ness would  have  been  ample.  The  more  flat  the  arch 
over  a  given  span,  the  less  the  thickness  required  for 
stability  of  form. 

No.  16.  A  very  heavy  work,  whether  we  look  to  the 
stability  of  form  or  to  the  ratio  of  2:>ressure  to  resisting 
power. 

Xo.  17.  This  Avork  has  ample  stability  of  form,  and  a 
pressure,  per  unit*  of  surface,  at  the  key,  nearly  equal  to 
the  limit  prescriljed  by  engineers.  The  least  thickness 
i-equired  for  stal>ility  of  form,  would  have  been  2'.rj3,  cor- 
responding to  7ir=1.0214;  but  this  would  have  caused  too 
great  a  pressui*e,  per  square  foot,  at  the  key. 

Xo.  18.  Nemours.  ?-=96'.394  ;  7^=1.033;  .5-14.2  X/. 
This  aicli  is  extremely  flat.  The  thickness  requir<'<l  for 
stal)ility  of  form,  is  only  I'.IS  (see  the  table  in  art.  145), 
corresponding  to  7^=1.0122.  This  thickness  should  be 
increftsed  until  the  mean  pressure  at  the  key  does  not 
exceed  yV  the  resisting  jiower,  or  until  the  ratio  in  the  last 
column  is  not  less  than  10. 


» 


GENERAL     REJIARKS.  401 

No.  19.  Turin.  r=l(jO'.20',  /r=1.029G;  .9=8.20/. 
The  tliickness  re(|uire(l  for  praeticaLility  (art.  145),  is 
about  3j  inclies  less  than  the  thickness  given.  The  pres- 
sure at  the  key  comes  up  to  the  limit. 


GENERAL  EEMAKKS  UPON  THE  DETERMINATION  OF  THE 
THRUST  AND  UPON  THE  THICKNESS  OF  THE  AP.CH  AT 
THE    KEY.  I 

151.  Given  the  span,  the  rise,  and,  approximately,  the 
load.  Assume  a  thickness  at  the  key  in  view  of  the 
strength  of  the  material,  the  probaljle  character  of  the 
workmanship,  the  load,  and  all  other  circumstances. 

If  the  arch  be  of  light  proportions,  look  for  its  thrust 
to  those  formuhe,  tables,  or  geometrical  methods,  whicli 
snppose  the  curve  of  pressure  to  pass  through  the  middle 
of  the  key  and  the  middle  of  the  weakest  joint,  or  joint  of 
greatest  thrust. 

On  the  other  hand,  if  the  arch  be  of  heavy  proportions, 
look  for  its  thrust  to  those  methods  which  suppose  the 
curve  of  pressure  to  start  one  third  the  length  of  the  joint 
from  the  extrados  of  the  key  and  to  pass  at  the  same  dis- 
tance from  the  intrados  of  the  weakest  joint. 

Ilavinor  found  the  thrust  in  cubic  feet,  divide  it  l>v  tlie 
length  of  the  vertical  joint,  or  thickness  of  the  arch  at  the 
key,  and  multiply  the  quotient  by  the  weight  of  a  cubic 
foot  of  the  material.  The  result  will  be  the  mean  pressure 
at  the  key  in  pounds.  If  this  mean  pressure  exceed  ^\ 
the  ultimate  resisting  power  of  the  material,  make  a  new 
supposition, — increase  the  thickness,  find  the  thrust  and 
mean  pressure  anew ;  and  so  on  until  the  results  seem  to 
be  satisfactory. 

We  cannot  draw  the  line  precisely  between  light  and 
and  heavy  arches.  Most  of  the  arches  of  table  I  are  light. 
With  one  exception,  their  thrusts  were  calculated  on  the 
first  supposition  mentioned  above.     The  exception  is  the 


402  TIIEORV     OF    THE    ARCH. 

Westminster  Bridge,  wliose  thrust  lias  Leeu  calculated  on 
tlie  second  sup])osition. 

We  liave  sliown,  art.  148,  tliat  in  the  worst  Y>ossil)le 
case,  that  of  a  semicircular  arch  surcharged  horizontally 
up  to  tlie  extrados  of  tlie  key,  we  are  at  liberty  to  determ- 
ine the  thrust  on  the  second  supposition  when  7^=1.14 
or  more,  or  when  the  span  is  14.29  times  the  thickness,  or 
less. 

We  may  therefore  adopt  this  value  of  K  as  the  divid- 
ing point  between  light  and  heavy  semicircular  arches. 

In  segmental  arches,  a  smaller  value  of  K  will  give 
the  limit,  art.  148.  Wlien  tlie  span  is  six  times  the  rise, 
the  limit  is  7i'==1.10;  and  this  may  be  taken  as  the  gene- 
ral limit  of  liglit  segmental  arches. 

In  elliptical  arches,  it  would  be  well,  perhaps,  to  raise 
the  limit ;  say  to  regard  all  as  light  in  which  the  rise  is 
five  times  the  thickness  at  the  key,  or  more. 

In  speaking  above  of  the  worst  possible  case,  we  have 
omitted  one  of  occasional  occurrence,  viz.,  that  in  which 
the  load  upon  the  reins  rises  higher  than  the  key,  the  lat- 
ter bein<4-  uncovered.  It  would  be  well  in  such  a  case  to 
raise  the  limit. 

152.  The  investigation  of  the  arch  indicated  in  the  pre- 
cedincT  article  will  not  in  all  cases  be  sufficient.  The  mean 
pressure  at  the  key  may  Ije  within  the  prescribed  limits, 
wliile  tlie  arch  is  impracticable  and  even  impossible.  We 
have  shown  briefly  how  to  investigate  this  case,  and  have 
given  the  limits  below  which  it  need  not  be  investigated. 

Very  light  arches,  only,  require  such  investigation. 
Their  thrust  should  always  be  determined  on  the  fii-st  sup- 
position of  art.  lol.  Having  obtained  a  thickness  which 
satisfies  all  the  conditions,  Ave  must,  if  the  arch  l)e  very 
light,  make  some  further  i)rovi.sion  for  the  change  of  form 
which  is  sure  to  take  place  after  the  removal  of  the 
center. 


I 


GENERAL    REMARKS.  403 

If  we  knew  tlie  compressibility  of  granite  and  otlier 
materials  nnder  a  given  pressure,  it  would  Ije  possible  to 
estimate  the  change  of  form  and  consecpiently  the  re([uired 
increase  of  thickness. 

In  the  absence  of  such  information,  we  can  only  study 
the  proportions  of  existing  arches.  There  are  two  ways  of 
making  the  increase. 

1st.  We  can  add  an  equal  thickness  to  the  arch 
throughout. 

2d.  We  can  gradually  increase  the  thickness  from  the 
key  to  the  si:)ring,  or  to  the  weakest  joint. 

The  first  method  will  be  the  best  wdien  the  arch  is 
covered  by  a  heavy  load  ;  the  second,  when  it  is  lightly 
covered  as  in  most  bridges. 

This  increase  need  not  in  general  exceed  50  per  cent, 
at  the  weakest  joint,  and  at  the  intermediate  joint  of  rup- 
ture, al)out  half  way  between  the  key  and  the  former,  it 
must  be  sufficient  to  keep  the  curve  of  pressure  within  the 
j)rescribed  limits. 

In  assuming  that  the  curve  of  pressure  shall  pass 
through  the  middle  of  the  key  and  the  middle  of  the  weak- 
est joint,  we  make,  in  fact,  an  impossible  supj^osition ;  for 
Avhen  we  reflect  upon  the  effect  of  pressure  and  compres- 
sion at  the  several  joints  between  the  key  and  the  regular 
lower  joint  of  rupture,  we  see  at  once  that  the  curve  of 
pressure  must  run  in  the  aggregate  about  equally  on  l)oth 
sides  of  the  central  line,  inclining  to  that  side,  generally 
the  upper,  on  which  the  particular  pressures  are  the  least. 

That  supposition  gives,  however,  the  proper  thrust  a 
little  in  excess ;  and  as  to  the  curve  of  pressure  we  have 
only  to  suj^pose  it  lowered  or  changed  until  the  proper 
disposition  is  attained.  Here  we  see  the  advantage  of  giv- 
ing to  the  very  light  arch  a  gradual  increase  of  thickness 
on  both  sides  of  the  key.  We  thus  attain  room  for  the 
curve  of  pressure,  and  make  practicable  perhaps  what 
might  otherwise  have  been  an  impossible  arch. 


404  TnEORY     OF    THE    ARCn. 

This  increase  aliove  the  lower  joint  of  rnpture  is  only 
useful  when  the  arch  is  inclined  to  the  third  mode  of  rup- 
ture ;  and,  instead  of  a  gradual  and  uniform  augmentation 
from  tlu^  key  to  that  joint,  it  would  perhaj^s  be  better  to 
increase  the  thickness  more  rapidly  to  the  intermediate 
joint  of  rupture,  then  more  slowly  or  not  at  all  to  the 
lower  joint. 

The  thickness  at  that  intermediate  joint  must  be  more 
than  equal  to  a  constant  thickness  lai'ge  enough  to  secure 
the  practicability  of  the  work. 

The  magazine  or  roof-covered  arch  is  not  inclined  to 
the  third  mode  of  ru|)ture.  We  may  always  take  its 
thrust,  including  the  effect  of  surcharge,  from  table  FF. 


GE03IETRICAL    METHODS. 

153.  Poncelet  has  mven  a  cfeometrical  solution  of  the 
ultimate  thrust,  No.  12  ^Memorial  de  TOfficier  du  Genie; 
and  his  method  might  be  modified,  with  little  change,  so 
as  to  give  the  actual  tlirust  of  the  arch.  We  follow  his 
method,  in  its  first  stages ;  but  by  introducing  some  of  the 
properties  of  the  curve  of  pressure,  we  have  been  able  to 
make  a  more  simple,  direct,  and  comprehensive  solution. 
In  one  or  two  hours,  a  person  a  little  familiar  with  the  ge- 
ometrical method,  can  lay  down  on  paper,  and  verify,  all 
the  elements  of  the  most  comi)lex  case.  This  method  is 
entirely  in(le})endent  of  all  particulars,  and  is  consecpiently 
especially  useful  when  irregularities  of  outline  or  construc- 
tion ])lace  the  arch  almost  beyond  the  reach  of  calculation. 

The  method  is  the  same  in  all  cases;  but  to  explain  it 
more  fully,  we  shall  apj)ly  it  to  loaded  and  unloaded  full 
circle  and  segmental  arches. 

Allusion  may  be  made  occasionally  to  letters  and  lines 
which,  to  avoid  confusion,  have   not   been  wiitten   and 


GEOMETRICAL    METHOD.  405 

traced  on  tlie  dingranis ;  Init  the  i)laces  of  sneli  Icttcis  and 
lilies  can  not  be  mistaken. 

A  brief  demonstration  or  justification  of  the  several 
steps  will  be  given. 


THE   UNLOADED    AT?CH. 

154.  Let  figure  48,  plate  XIL,  represent  the  proposed 
arch,  of  any  intrados  and  extrados.  The  semi-arch  is  lim- 
ited by  the  indefinite  horizontal,  C  E,  of  the  springino- 
line  ;  the  indefinite  vertical,  C K,  passing  through  the  cen- 
ter of  the  key  ;  the  outline  i  i^  i^...  i\  of  the  intrados  ; 
and  the  outline  6  ^i  ^2  •  •  •  ^c  of  the  extrados. 

I.  Point  of  application  of  tlie  thrust.  This  must  be 
between  c  and  c\  points  which  divide  the  joint  of  the  key 
into  three  equal  parts.  In  heavy  arches,  loaded  or  un- 
loaded, we  generally  assume  the  upper  limit,  e,  as  the 
point  of  application  ;  in  light  arches,  so  loaded  as  to  be  in 
danger  of  the  third  mode  of  rupture,  the  crown  rising,  we 
generally  assume  c',  the  low-er  limit,  as  the  point  of  appli- 
cation. 

The  same  lower  limit  should  be  assumed  in  the  gothic 
arch. 

II.  Nearest  approacli  of  tlie  curve  of  presmre  to  the 
intrados  and  to  tlie  extrados.  Draw^  the  curves  e',  e'l,  d^ . . . 
(^'a ;  ^5  ^1,  <^2  •  •  •  <"&,  dividing  the  joints  of  the  arch  into  three 
equal  parts.  The  curve  of  pressure  must  not  pass  outside 
of  these  limits  between  the  key  and  the  joint  of  greatest 
thrust. 


III.  Division  of  the  arch  into  fictitious  voussoirs. 
Divide  the  semi-arch  into  4,  6,  8,  or  10  segments,  accord- 
ing to  its  absolute  size,  and  to  the  degree  of  accuracy 


406  THEORY    OF    THE    ARCH. 

which  we  wish  to  attain.  Let  these  segments  have  a  com- 
mon altitude,  so  chosen  as  to  effect  the  desired  division. 
Make  the  division  as  follows :  from  f,  with  the  common 
altitude  as  a  constant  radius,  describe  a  small  arc  and 
draw  the  joint  i'l  e^  tangent  thereto.  The  joints  are  gene- 
rally pei'pendicular  to  the  intrados.  Next,  from  e^  as  a 
center  and  with  the  same  radius,  describe  another  arc,  and 
draw  the  joint  4  Ci  ^2  ^2  tangent  thereto . . .  and  so  on  to 
the  springing  line.  We  need  not  waste  time  in  trying  to 
make  this  division  come  out  exact.  The  last  segment,  %. 
50,  may  have  an  altitude,  e^i\  a  little  more  or  less  than 
the  constant^  a  p. 

IV.  Partial  and  total  areas  of  the  fictitious  voiissoirs. 
Draw  the  parallels  i  d^  to  e  /i,  meeting  e^  i^  prolonged  at 
d^ ;  /j  di  to  <?i  /g  meeting  <?2  io  prolonged  at  c/2 .  .  .  and  so  on 
to  the  springing  line.  If  the  last  altitude,  ^5^^,  fig.  50, 
comes  out  unecpial,  lay  off  ^;  a  on  j)  4,  prolonged  if  neces- 
sary, p  a  Toeing  the  constant  altitude,  which  we  will  desig- 
nate l)y  a.  Draw  /g  a  parallel  to  e^  ?c ;  then  e^  d^  parallel 
to  a  a\  and  <?5  d\  parallel  to  a  e^.  The  surfoce  '4  i^  e^  e^  is 
measured  by  aX^d^  d\.  The  area  of  any  other  segment, 
as  i.2  is  fa ^2,  i-"^  measured  by  iaxd^  e^.  It  will  be  seen  that 
we  regard  the  right  lines  supposed  to  connect  i  ?i,  ?'i  4 . . . 
e  ^1,  e^  62,  <fec.,  as  representing  the  outlines  of  the  arch. 
This  supposition  is  generally,  in  practice,  favorable  to  sta- 
bility. 

On  the  vertical  C K^  from  the  assumed  point  of  appli- 
catii.n  of  the  thrust,  which,  in  fig.  48,  is  c,  the  upper  limit, 
lay  off  e  s^=.\d^  e^,  s^  s^—\d.2  e^,  and  so  on  to  the  .springing 
line.  Then  the  partial  segments  will  be  measured  as  fol- 
lows, viz.:  the  surface  i  i^e^e  by  ^^X^'-^i,  surface  i^iie^Cx 
by  a  X  s^  .5-2,  itc. ;  and  the  total  segments,  i  4  e^  e  by  a  X  c  S2, 
iis^s^  Ijy  a  ><<''%  *^<^'  Area  of  the  entire  semi-arch  = 
a  X  c  'S'e,  or  a  X  c  -Ss, ...  as  the  segments  may  be  G,  8,  etc.,  in 
number.     If  the  line  0 -Se  or  c  .Vg .  .  .  is  f<>uii<l  to  extend  too 


GEOMETRICAL    METHOD.  407 

fur  on  the  drawing  for  convenience,  all  tbe  distances  c  s^^ 
6'!  6^2.  .  .■%  S(,  may  be  reduced  by  one  half,  in  Avhich  case  the 
areas  of  the  several  segments  will  l)e  measnredby  c  s^^  -S'l  -s^ 
.  .  .G-%,  etc.,  multiplied  by  2«;  that  is,  by  a  into  the  recip- 
rocal of  the  fraction  of  reduction. 

Di'aw  the  indefinite  horizontals  s^  Wi,  s^^  ii^.  .  .•Vc  "c- 

Y.  Center-^  of  gravity  of  the  2^(iJ'tial  sefjinents  or  voii-s- 
soIj's.  Of  any  quadrilateral,  fig.  49,  draw  the  diagonals 
intersecting  at  7n  ;  lay  off  c^  a—7)i  4,  i\p=m  e^,  and  mark  <\ 
the  middle  of  e^  /g,  and  n^  the  middle  of  i^  e^ ;  join  a  n^  cp; 
their  intersection,  ^4,  is  the  center  of  gravity  of  the  figure. 
This  point  is  sometimes  given  more  conveniently  by  taking 
c  o^—^cp^  or  n  6*4=4  '^  '^'-  ^^^  *^^^  manner,  or  in  any  other 
more  expeditious  manner,  determine  the  centers  of  grav- 
ity 6>i,  0.2,  Og,  tfec,  of  all  the  segments. 

YI.  Centers  of  gravity  of  the  total  -segments. 

Draw  a  horizontal  through  some  point,  K\  about  as 
far  below  the  point  of  application  of  the  thrust  -as  the  ex- 
treme center,  0^,  is  distant  from  the  vertical  C  c. 

Project  vertically  on  that  horizontal  the  centers  0^,0^^ 
03, . . .  respectively  at  o\,  0'^,  o'-^,  tfec.  Prolong  o\  c  until  it 
intersects  the  horizontal  through  s^  at  m^ ;  thence  draw 
???!  m.,  parallel  to  o'a  e,  meeting  s.i  v^  at  Wc,;  thence  ^n^  nta^ 
parallel  to  o'g  c,  meeting  .53  v-^  at  w?8,  and  so  on  to  the 
springing  line. 

Project  Oi  vertically  on  the  horizontal  en-,  at  n^.  Lay 
off  c  K—c  Iv  ;  draw  Kn.,  parallel  to  m.  c  meeting  c  n-  at 
Uo  ;  Kih  parallel  to  Wg  c^  meeting  c  nj  at  ??3  ;  and  so  on  to 
??6,  corresponding  to  w?6,  a^tl  to  the  springing  line. 

The  points,  n^,  ih,  ih,  iu,  <fec.,  on  the  direction  of  the 

horizontal  thrust,  are  directly  over  the  centers  of  gravity 

of  the  total  segments,  ih  corresponding  to  i  h  <?2  <?,  or  seg- 

•    ments  1  and  2  combined;  /^3  to  segments  1,  2,  and  3  com- 

15 


408  THEORY     OF    TUE    ARCH. 

Lined,  etc. ;  Jh  i^  *'^*^'i"  ^^^*'  ^"'■uter  of  gravity  of  the  wliole 
semi-arcli.  {a) 

YU.  The  ad  tied  tliru-st.  Let  ll^^  determine  tlie  hori- 
zontal force,  which,  acting  at  (\  shall  hold  the  fii'st  segment 
in  equilibrimn  on  c\,iic\=^ixe^.  ihe\  is  oLvionsly  the 
direction  of  the  resultant  of  this  force  and  of  the  weight  of 
the  segment,  represented  by  c  s^.  Draw  c  t^  jiarallel  to 
f'l  ;?i,  meeting  .9i  ?«i  at  /j/  -<;-i  ^i  is  the  force  required.  In 
like  manner  draw  c  f.^  parallel  to  c-'o  ih,  meeting  .%  m^  at  A,  ; 
c  t-i  parallel  to  c\  i^  meeting  ^3  ???3  at  f^ :  and  so  on,  until 
the  ordinates  s^  /j,  .s-2  A,,  ttc,  having  attained  their  maxi- 
mum, begin  to  diminish.  Draw  a  vertical,  v^  V(,,  tangent 
to  the  curve  t^  U  Ui^...  If  ^3  be  the  point  of  tangency, 
and  /=the  greatest  horizontal  ordinate  of  the  curve  (in 
this  case  t—-%  ^3),  the  actual  thrust  expressed  in  cuTjic  feet, 
or  cubic  units,  if  other  than  feet  be  used,  will  1)6  F=aXt. 
We  must  not  forget  to  double  ei  if  the  lines  c  .^1,  Sy  .5'2,  <fec., 
have  been  reduced  by  \  . .  See  IV. 

VIII.  The  ewve  of  pressure.  Mark  v^,  ?^,  ?/«,. .  .Vr,, 
the  intei-sections  of  the  vertical  just  mentioned  with  s^  rn^, 
Som^,  Ss  rris,. .  .s^  m^.  Draw  71^  .Ti  parallel  to  v^  c,  meeting 
the  joint  i^  <?i  at  x^ ;  n^_  a\  parallel  to  Vo  <;  meeting  4  ^t  at  x.. 
. .  .??6a'6 parallel  to  v^o^  meeting  the  springing  line  at  x^. 


(a)  Let  «,,  «a,  i!3,  «fcc.,=rtlie  areas  of  the  combined  segments,  1  ;  1  A  2  ;  1 , 2,  A  n,  Ao. 
"  TWi.jnj,  Wj,  <fcc.  =  moni't3  on  6'   "  "  " 

Tlien  «j— »,=area  of  the  2d  segment;  «3— «2=area  of  3d  segment,  etc. 
"    w'a  —  iW=mom't         "         "  Wa  —  w*j  =  mom't "         "  " 

We  liave,  by  con^'tniclion, 

A'o'i  X  cui      vii 


•A"  :  K  u\  ::  ra^  ;  *■,>/(,  =  - 


cA"  cJC 

A'o's  X  *j/!j        Jflj  — Wi 


<A"  :  K'o'-i  ::  x^s.^  :  g.^w^— #iWi^ 

Hence,  «2»ns  =  -^.,.   In  like  manner /<3W3=     .,  »te.  We  also  have  c«j  :  s.,//in ::  cA';  cxgzr 
cA  r/i  • 

fA'x^jOTj        Wj       ^      ...  "Is  III*      , 

= In  like  manner, eKa  =  — ,  cu,=  —  ,  &c. 


GEOMETRICAL    METHOD.  409 

The  curve  of  pressure  passes  tlirono-li  tli(.'  ])(>liit<  c,  ,/■„ 
tT.2,  ^T3 .  .  . .Tg.  (/>)  It  cannot  coine  within  the  inferior  limit, 
^''t  ^'\}  t*'25<fec.;  bnt  it  will  generally  approach  tlie  outer  limit 
above  the  springing  line,  if  the  arch  be  lull  ciicle. 

If  the  curve  of  pressure  runs  above  the  u])per  limit 
immediately  after  leaving  the  key,  we  must  regard  the 
assumed  point  of  application  of  the  thrust  as  impracticable. 
A  lower  point  must  be  taken.  If  the  curve  corresponding 
to  this  last  point  still  pass  l)eyond  the  outer  limit  hefore 
toucJting  tlie  imiei\  a  still  lower  point  must  l^e  taken.  If 
the  curve  of  pressure,  starting  from  the  upper  limit  at  the 
key,  fall  immediately  below  the  upper  limit  cui've,  we  need 
not  investigate  the  thrust  or  the  point  of  application  any 
further. 

IX. .  Chancjiiuj  tlie  point  of  application  of  the  tlivust. 
Suppose  we  have  obtained,  as  above,  the  actual  thrust  and 
the  curve  of  pressure  corresponding  to  any  point  of  appli- 
cation, say  c,  ic=^%ie^  and  wish  to  determine  the  variations 
in  both,  consecpient  upon  a  change  to  any  other  point  of 
application,  say  g\  ic  ^=-^ie. 

Project  the  points  ??i,  'Uo,  th^^^h^  etc.,  already  determined, 
vertically  on  the  horizontal  passing  through  the  new  ]X)int 
of  application,  c'.  Mark  these  projections  as  n\^  n'.,  ?/g, 
etc.  Determine  a  new  curve  of  thrusts  (VII.)  by  drawing 
ct\  parallel  to  c\  n\,  meeting  6\  m^^  at  t\  ;  c/'a  parallel  to 
C.2  'n'^t  meeting  -s.^  m^^  at  t'.^i  etc. 

Draw  a  vertical  tangent  to  the  new  curve  ?!'i,  t\^  t'-.^,  etc., 
and  mark  its  intersections,  u\  with  Si  m^^  iu_  with  s.^  rn.^^ 
etc. 

The  greatest  horizontal  ordinate  of  this  curve  =yi  u\  — 


{!))  We  have  c.s3=:the  vertionl  force  due  to  tl>e  total  segment  i  is  rs  e;  .13  n3  =  ihc 
actual  thrust  or  constant  horizonlal  force.  Tlie  diagonal  V3  c  is  the  direction  of  th. 
resultant ;  113  is  one  point  in  it. 

The  same  considerations  apply  to  the  other  points. 


410  THEORY    OF    TUE    ARCH, 

•v..  k'o,  etc.,  is  the  actual  tbrust  imtler  the  coiulitions  im- 
posed. 

Determine  the  new  curve  of  pressure  liy  drawing  n'l 
.r'l,  parallel  to  ?/'i  c,  meeting  i\  e^  at  .v\\  n\  x\  parallel  to 
u\_  e,  meeting  /g  ^2  ^^  '^"'s  ^^^^^  ^^  <^>»?  to  the  spriuiriiig  line. 
These  changes  are  made  in  a  few  minutes  by  a  repetition 

of  steps  already  explained.     The  points  .5-1,  s^^  s^ Wj,  m^i 

etc.,  are  laid  off  once  for  all.  To  avoid  confusing  the  dia- 
o-ram,  the  new  curves  of  thrust  and  of  pressure,  ai"e  not 
indicated  in  fig.  48.  In  like  manner,  any  number  of  points 
of  application  may  be  assumed  and  their  corresponding 
thrusts  and  curves  determined.  But  it  will  rarely  be 
necessary  to  assume  more  than  two  or  three  points. 

X.  Limits  of  possible  and  pnicticalh  an-lics.  If  we 
suppose  the  curve  e',  c\,  c'o,  •  •  •  which  marks  the  nearest 
approach  of  the  curve  of  pressure  to  the  iutrados,  to  be 
the  intrados  itself,  and,  assuming  /,  the  lowest  point  of  the 
kev,  as  the  point  of  application,  find  the  resulting  curve  of 
pressure  to  pass  outside  of  the  extrados  between  the  key 
and  the  joint  of  greatest  thrust,  we  know  the  arch  to  be 
impossible^  whatever  the  resisting  power  of  the  materials. 

If  the  curve  of  pressure  starting  from  c ,  the  lower  limit, 
and  touching  that  limit  again  at  the  joint  of  greatest 
thrust,  is  f(jund,  at  some  intermediate  joint,  to  pass  beyond 
the  upper  or  outer  limit,  we  regard  the  arch  as  imprac- 
ticalde. 

The  curve  of  pressure  can  not,  in  such  a  case,  be  con- 
fined within  two  other  curves  wliicli  divide  the  joints  into 
three  equal  parts,  and  the  joints  of  the  arch  must  open  at 
the  intrados,  or  extrados,  or  botli. 

XI.  IJie  casf'  of  two  ime'judl  nn-lieSyOr  arches  heariiuj 
unequal  loacU^  n'itli  a  vouonoii  ].•<  ij. 

Figure  48  representing  the  seini-;uvh  of  least  tlirnst,  let 
.s^  ?^"j=:the   thrust   of  the  t>ther  part.     First,  to   test  the 


GEOMETRICAL    IMETIIOD.  411 

possibility  of  the  smaller  seini-arcli,  assume  /  as  the  point 
of  application ;  on  a  horizontal  passing  through  i  jn-oject 
??i,  v?25  ^'35  'iUi  etc.,  already  determined  in  relation  to  any 
other  point  of  application.  Call  these  projections  ?i"i,  n^^ 
n"s,  etc.  Continue  the  vertical  tlirough  n'^  meeting  -p^  111.. 
at  i/Z'a,  .%  W3  at  i/'g,  etc.  Construct  the  curve  of  pressure 
in  the  usual  manner,  by  drawing  n\  x\  parallel  to  v'\  0 
meeting  i^  e^  at  a^\  ;  n\^  x\  parallel  to  u\  c  meeting  ?2 <?.2 
at  x'\,  etc.  If  this  curve  pass  outside  of  the  extrados, 
the  arch  is  impossible. 

Second,  to  test  the  practicability  of  the  semi-arcli, 
assume  c\  'ic'=^^ie,  as  the  point  of  application.  On 
the  horizontal  through  c  project  vertically  the  points 
7?i  at  ?i'i,  ??.2  at  ^'2,  »3  at  ^'3,  etc.,  and  determine  points  in 
the  curve  of  pressure  by  drawing,  from  these  last  points, 
parallels  n\  x\  to  u'\  c,  meeting  ?\  e^  at  a?'i,  ^z'g  ^%  to  ii\  c, 
etc.  If  the  curve  thus  determined,  pass  beyond  the  outer 
limit,  ^,  ^1,  ^2,  ^3,  . . .  etc.,  the  arch  is  impracticable.  At  the 
point  where  it  crosses  that  limit,  it  will  be  necessary  to 
bemn  to  increase  the  thickness  of  the  arch. 

XII.  Direct  ion  and  magnitude  of  tlie  iwessiire  xiimi  the 
several  joints.  As  the  actual  thrust,  .^3  ^3,  %.  48,  is  the 
constant  horizontal  force  acting  upon  all  the  joints,  while 
c  .9i,  c  s^,  G  s-i .  .  etc.,  represent  the  surfaces  or  weights  of  the 
total  segments,  1 ;  1  and  2  ;  1,  2,  and  3,  etc.,  it  is  obvious 
that  the  diagonals,  n^c,  Vo/\  u^c,  v.c,  etc.,  represent,  both  in 
magnitude  and  direction,  the  resultants  which  press  upon 
the  several  joints ;  viz.,  u^c  on  2^1,  UiC  on  V2,  ^^s^  on  V3,  etc. 

If  the  angle  between  any  one  of  these  resultants  and 
the  corresponding  joint  should  he  less  than  the  comple- 
ment of  the  angle  of  friction,  say  less  than  60°,  rupture 
may  take  place  by  sliding.  Lay  off  these  diagonals  from 
the  same  point  on  any  line,  as  c  'u\,  fig.  54  ;  make  uv=ie; 
continue  the  line  c' v.  The  intercepted  ordinates,  -?/.>?•>, 
^^'3^3,  etc.,  will  give  the  required  lengths  of  joint  on  the 
condition  that  these  shall  be  proportional  to  the  resultants. 


412  THEORY    OF    THE    ARCH. 

XIII.  TliC  tlirast  hy  didiiKj.  The  horizontal  force  P' 
necessary  to  hold  any  surface  S  in  eqnilil:)rinm  on  a  joint 
making  the  angle  v  with  a  vertical  is,  a  being  the  angle  of 
friction,  P'=:*S'xcotang.  (rt-f-r).  Assume  ^/=:  30°.  Then 
P'=>S'Xtang.(GO°-t'). 

Draw  the  line  C l\  iig.  48,  G0°  from  a  vertical;  draw 
c  ^"i— angle  s^  c^"i=ang.  v  6'/i— 60°  — t',  meeting  s^  ni^  ^^t 
f'l'-,  <?^"2— ang.  Si  ct"2=^f^  Oi-i^  meeting  <<?2  ?>?2  at  t"o,  and 
so  on. 

The  greatest  horizontal  ordinate  of  the  curve  t"i  t'^t"^...^ 
is  the  thrust  by  slidinu'.  The  corresponding  joint  of 
greatest  thrust  is  generally  inclined  about  oO''  to  the  ver- 
tical. It  would  be  well  to  interpolate  one  or  two  joints 
near  30°,  with  altitudes =^^^ 

Should  the  thrust  due  to  slidinoj  exceed  that  due  to 
I'otation,  the  former  will  be  the  actual  thrust.  The  point 
of  application  may  be  assumed,  generally,  as  near  the  cen- 
ter of  the  key,  and  the  curve  of  pressure  may  be  determ- 
ined precisely  as  in  XL 

XIV.  Thichiess  of  Pier.  Continue  i^  a'g,  correspond- 
ing to  the  springing  line,  fig.  48,  to  the  base,  x-,.  A  p— 
I-  A  x-i  will,  if  the  height  of  the  pier  be  small,  be  very 
nearly  the  thickness  required  on  the  condition  that  the 
curve  of  pressure  shall  meet  the  Ijase  at  one  third  its 
length  from  the  exterior  edge.  2xAx-,—AP^^  is  the 
thickness  required  in  order  tliat  the  curve  of  pressure 
shall  pass  through  the  middle  of  the  base.  The  exact 
distance  may  be  obtained  most  easily  by  a  curve  of 
errors.  Let  ,T=the  distance  between  the  exterior  edge  of 
the  base  of  the  j)ier  and  the  intersection  of  this  base  with 
the  curve  of  pressure;  t.=tlie  required  thickness  of  pier; 
^y=:the  ratio  of  x  to  <?;  so  that  a?=^;X<?.  2^  is  usually 
assumed  at  one  thiitl,  sometimes  as  lai'ge  as  one  half 
Having  already  obtained  the  ju'ovisioiial  thickness  ^1 /?i, 
mark  /•,  such  that  i\  X'^^>y.i\A.     If  ^;=:i,  ;*i  .Tv^l.rj^l. 


GEOMETRICAL    METHOD.  41;> 

ABt^  being  too  large,  lay  off  tlie  error  B^  i\  above  7/,  ; 
this  gives /i.  Assume  AB.^  as  another  thickness.  Deter- 
mine the  distance  BojJ-;^  precisely  as  if  the  surface  AB.>  E.^ 
■U  were  a  segment  of  the  arch,  such  tliat  aX^B.^'J-  — 
A  B.2  X  B.2  E2.  Lay  off  -L  /j^  cl-i  from  s^  i-eaching  to  '-sv, 
draw  the  horizontal  s~  v-.  Project  the  center  of  gravity  of 
A  B^  E<2  /g  on  K'o\  at  d-^ ;  draw  Wg  m-;  parallel  to  d-/.\ 
meeting  Si  Vj  at  w^ ;  draw  lui-i  parallel  to  m-jr^  thence 
n-;  iTs  parallel  to  'ft^c.  This  gives  cCq.  Mark  r^  such  tliat 
;'2  tTs=:^iX^'2-4i  ^i^tl,  AB^  being  too  small  a  thickness,  lay 
off  the  error,  B^  i\^  below  B^ . .  giving /g-  Join/^/a  meet- 
ing the  line  of  the  base  at  B.  ^^  is  the  thickness 
required,  very  nearly.  Verify  this  by  assuming  AB  as 
the  thickness ;  determine  .97,  w^,  n^^  and  x^  anew ;  this  will 
at  least  give  a  new  point  in  the  curve  of  errors  very  near 
the  base.  Connect  this  point  with  the  more  distantyi  or 
f^  and  mark  the  new  intersection  with  the  Ijase.  Further 
trial  will  rarely  be  necessary. 

If  we  wish  to  cover  the  pier  above  the  springing  line, 
by  a  mass  of  masonry  sloj)iug  gradually  to  the  extrados  of 
the  arch,  we  must  take  into  account  this  new  surface  and 
its  moment  as  a  part  of  the  pier,  adding  its  surface  to  c'-9j, 
projecting  its  center  of  gravity  on  It'o'r,  and  so  on,  in  the 
usual  order  to  the  intersection  of  the  curve  of  pressure 
with  the  base. 

XV.  The ])ressure  per  unit  of  surface.  The  pressure 
at  the  key  being  F^tXci^  the  mean  pressui'e  at  that  joint, 

d  representing  its  length,  is, ;  multiplying  this  hy  u\ 

d 

the  weight  of  a  cubic  unit,  we  have  the  pressure  in  pounds  = 

TF=:- — ^ .     The  mean  pressure  at  any  other  joint,  as 

d 

U  <^4,  ns;.  48,  IS  ~ .     ilie  mean  pressure  should  not 

exceed  -^\  the  ultimate  resisting  poAver  of  the  material. 


414  THEORY    OF    THE    ARCH. 

The  greatest  clanger  will  be  at  the  joints  of  rupture,  so 
called,  where  the  curve  of  pressure  approaches  nearest  to 
the  outlines  of  the  ai'ch.  AVe  here  use  the  pressure  or 
resultant  itself,  not  its  normal  component. 

AVe  might,  witli  more  accuracy,  use  the  latter,  if  we 
took  notice  also  of  the  com])onent  parallel  to  the  joint 
wliich  generally  tends  to  cause  the  rupture  which  we  wish 
to  prevent. 

XVI.  Increase  of  tlie  arch  Idow  the  joint  of  greatest 
til  rust.  The  proper  thickness  of  the  full  circle  or  semi- 
elliptical  ai'ch  at  the  springing  line,  we  can  best  estimate 
after  laying  off  the  curve  of  pressure  down  to  the  joint 
above. 

The  last  exterior  segment  must  be  at  least  equal  to  one 
third  the  length  of  the  joint. 


THE    LOADED    ARCH. 

155.  Let  fig.  51  represent  the  proposed  arch,  with  any 
intrados  and  extrados,  and  sustaining  any  load  rising  to 
Pi  Pu  Pit  etc.  As  to  the  point  of  application  and  the  situ- 
ation or  range  of  the  curve  of  pressure  (see  I.  and  II.  of 
the  preceding  article. 

III.  Division  of  the  arvli  into  fictitious  voussoirs  or 
segments.  Divide  the  semi-arch  into  4,  0,  8,  or  10  seg; 
ments,  with  a  common  altitude  drawn  from  the  upper 
angle  of  each  perpendicular  to  the  opposite  side  or  joint, 
precisely  as  in  the  preceding  article.  From  the  points  of 
the  extrados  thus  determined,  I'aise  the  verticals  ^^>,  e^jh 
e>2pi^  etc.,  through  the  mass  of  the  surcharge,  suppo^ied  to 
have  the  density  of  the  arch  pro])er.  Bear  in  mind  the 
last  part  of  III.,  154. 

IV.  Partial  and  total  areas.  Di'aw  the  j)arallels,  /  d^ 
to  e  /„  meeting  <?,  /j  at  d^  ;  i^  d^  to  e^  ?2,  meeting  ^2  H  at  ^/g. 


GEOMETRICAL    METHOD.  415 

mid  SO  on  to  the  springing  line.  Draw  tlie  parallels  p  l\ 
to  e2h^  meeting  e^ih  ^^  hj  tlienee  l\  d\  to  e^  6,  meeting  ix 
<?i  at  d\ ;  j>x  li\  to  e^lh^  meeting  ^27^2  ^'^  ^'2?  tlience  lx\  fhi  to 
^1  e<2,^  meeting  ?2  e.^  at  d\^  and  so  on,  to  the  springing  line. 

On  the  vertical  Cc  passing  tlirongh  the  middle  of  the 
key,  lieginning  at  <?,  the  assnmed  point  of  application  of 
the  thrnst,  lay  off  c  8\=^^dx  d\,  s-^  •'^2=i<^4  *^^'27  ^nd  so  on  to 
the  springing  line.  Then,  area  of  the  1st  segment  includ- 
ing its  load— «XC'^'i ;  area  of  2d  segment=rt  X-'?!  -"^o,  (^"^c  ; 
1st  and  2d  combined=(r/  X c  s^ ;  1st,  2d,  and  3d  combined^ 
a  X  c  s^^  etc. 

In  fig.  51,  for  want  of  room,  we  have  reduced  these  dis- 
tances, csi^  s^-s^i  s^s^^  etc,  by  one  half,  so  that,  in  the  for- 
mulae just  given,  a  must  be  replaced  by  2«. 

Draw  the  indefinite  horizontals  s-^,  v-^,  s^  7^2,  •-'^'3  ^zi  etc. 

V.  Centers  of  gravity  of  tlie  2^cirticd  segonents.  Determ- 
ine 0',  fig.  52,  the  center  of  gravity  of  any  segment  of 
the  arch  proper  (see  V.  154)  ;  and  0,  the  center  of  gravity 
of  the  superincumbent  load.  Drawing  perpendiculars  da\ 
0  a,  to  the  line  0  o\  make  d  a=e^  d'^,  0  a=ds  e^ ;  join  a  a 
meeting  0  d  at  o^.  This  gives  the  center  of  gravity,  ^3,  of 
the  3d  segment  and  its  load.  («) 

In  like  manner  the  other  centers,  o^,  O2,  ^4,  ^0,  e^^'-?  "''''^y 
be  found. 

All  the  remaining  parts  of  the  construction  are  pre- 
cisely the  same  as  in  the  preceding  article,  except — 

XIV.  TMchuss  of  Pier.  A  B  being  the  line  of  the 
base,  add  to  the  semi-arch  that  part  of  the  pier  which 
underlies  the  assumed  thickness  at  the  springing  line. 
Lay  off  .55  -^c  representing  its  surface ;  project  its  ceiiter 
of  gravity  at  ^'g,  and  determine  in  the  usual  manner  the 
corresponding  points,  Wg,  ??6  and  finally  x^. 

(fl)  The  parallels  o  a,  o  «,  are  not  necessarily  perpendicular  to  o  o'  but  may  be 
drawn  in  any  other  more  convenient  direction. 


416  THEORY    OF    THE    AKCII. 

]\[n]-k  7'i,  rj  ,rc=-l-.ro^4.  On  the  line  e^B^^  above  B^  if 
A  Bi  ]je  too  large,  below  if  too  small^  lay  off  the  error 
Bi  fi=B^  Ti.  Assume  another  thickness,  say  A  r^^  and, 
corresponding  to  the  addition  thus  made  to  the  ])ier,  B^ 
'^'xlhpbi  determine  the  suite  of  points  6>V,  -^7,  ^?7,  x^  Lay  off 
.7';  ^2=^  A  x~,  and  7\fo^=/\  n,  the  corresponding  error,  /g 
■would  have  been  laid  oft' above  the  base  if  the  supjiosed 
thickness  had  been  too  great. 

If  A  Bi  had  been  too  great,  the  line  r^  2^  would  have 
been  on  the  right  of  B^^  s<i  s-i  laid  oft*  from  Sq  doMu wards, 
)n(,m-  drawn  backwards,  parallel  to  e  o\.  Continue /1/2, 
meeting  the  line  of  the  base  at  B.  A  B  \<,  the  recpiired 
thickness,  nearly,  corresponding  to  a'^-j^?,  or  7>=:^  (see 
XIV.,  art.  154).  In  like  manner  the  thickness  required  for 
any  other  value  of^),  may  be  determined.     In  general, 

Pi  Pa 

•'^'c  i\  =  -. XoOf^A,  (Vj  }\ = -^ —  X .v-  A . 

1-2^  \—p 


SEG3IENTAL    AECIIES. 

15G.  These  may  be  treated  precisely  as  semicirculni'  or 
semi-elHptical  arches..  Light  arches,  however,  loaded  or 
unloaded,  admit  of  a  more  simple  metliod,  of  Avhich  the 
only  error  consists  of  a  slight  exaggeration  of  the  thrust, 
and  of  the  dangers  to  which  the  structure  is  exposed. 

This  method  is  recommended  for  universal  use  in  light 
arches  of  large  sj)an.  We  have  taken  for  illustration,  an 
arch  Avhich  Capt.  Meigs  is  now  building  on  the  Washing- 
t<»n  a(|U(Mliic't ;  fig.  53  representing  the  arch  as  unL)aded; 
fig.  54,  the  same  \vith  its  final  load.  Tlie  s])an  is  <S'=220 
feet;  the  rise  /=57'.2GG;  the  thickness,  at  the  key  (l—\ 
feet,  at  tlie  spring  ^/'=6  feet;  the  semi-angle  of  the  open- 
ing v=if)h° ;  the  L)ad,  supposed  to  rise  to  a  horizontal,  and 
to  have  the  density  of  tlic  aich,  is  12  feet  dee])  over  the 
key, /=12'. 


GEOMETRICAL    METHOD.  417 

157.  Tlie  iinlocuJed  'Segmental  arcli^  fig.  53. 

Divide  the  arcli  l)y  vertical  lines  y'l  ^j,  ^  ^2,  U  ^sj  t^te.,  at 
equal  horizontal  distances  apart,  into  four,  six,  or  more  seg- 
ments, according  to  the  extent  of  tlie  angle  and  to  the 
absolute  size  of  the  sti-ucture.  Midway  Ijetween  these 
joints  lay  off,  through  tlie  arch,  the  verticals  d^  d\,  d^  d'^, 
ds  t/g,  etc. 

Points  in  the  direction  of  all  these  verticals  may  be 
obtained  at  once  by  dividing  the  half-span,  or  any  e(pial 
and  parallel  line,  into  twice  as  many  equal  spaces  as  we 
make  segments  in  the  semi-arch. 

Calling  a  the  common  horizontal  distance  between  the 
joints  ?'i  e^  and  4  e.,^  etc.,  any  one  of  the  segments,  as  '4  u  e^e^, 
will  be  measured  by  « X  ^^4  d'4.  At  the  same  time,  these 
midway  mean  depths  of  the  segments  pass  very  nearly 
through  their  centers  of  gravity ;  Ave  assume  them  to  pass 
through  these  centers,  the  error  here,  too,  being  in  favor  of 
stability.  Project  vertically  on  the  horizontal  through 
IC.,  di  at  o'l,  d.2  at  o'.2,  d^^  at  o'^,  etc.  Trace  the  curves  c'  c\ 
c'2  c-'g . . . ,  c  Ci  C.2  <"3 . . .  dividing  the  actual  joints  (perpen- 
dicular to  the  intrados)  into  thi'ee  equal  paits.  Make 
c  6\^d^  d\,  6\  .s'2  =  d.2d'.2-,  etc.  All  the  remaining  steps  are 
precisely  as  in  art.  154. 

In  this  jjarticular  case  we  see  that  the  curve  of  pressure 
starting  (as  it  generally  may  in  the  unloaded  arch)  from 
the  upper  limit,  and  touching  the  lower  curve  at  the  joint 
of  greatest  thrust  betweei^  u  ^ncl  4,  passes  over  to  the 
upper  limit  again  at  the  springing  line.  The  arch  is  prac- 
ticable, and  that  is  all. 

All  the  error  we  commit  in  supposing  the  joints  to 
be  vertical,  is  that  of  neglecting  the  little  surface  c'^  d'  fo, 
which  tends  to  diminish  the  thrust,  and  adding  the  effect 
of  the  still  smaller  surftice  dc'd'/Q.  At  other  joints  the 
ei-ror  will  be  still  less. 

The  tJiichiess  of  pier  may  be  obtained  by  construction, 
as  already  explained,  or  by  the  formula^  of  art.  128;  in 
using  the  latter,  remember  that  n=^c  '>oX</,  m^n  xPih- 


■lis  THEORY     OF    THE     ARCH. 

ir»S.   The  loailed  ■9e(/raental  cdtIi^  fig.  54. 

Divide  the  senii-arcli  precisely  as  in  ir>7,  tlie  vertical 
joints  />,  /j  ^1,  /o  ^25  ^tc.,  being  at  equal  horizontal  distances 
apart,  and  the  broken  lines  c/j  (1\,  tl,  ^/'.,,  d^  <1\,  etc.,  nlid^vay 
between  the  former,  extending  from  the  chords  i  i\,  f\  Ui 
?2  /s,  etc.,  to  the  upper  surfjice  of  the  surcharge.  On  the 
horizontal  K'  o'r,  project  vertically  <]^  at  o'l,  (!<,_  at  o'o,  cl^  at 
o'g,  etc. ;  lay  oif  <?'  -S'^^^rJ^  (J\  ;  .9^  .«?2=^/o  /-/'o,  etc. ;  extend  o\  c 
to  w?i,  thence  draw  ?/^i  ui.^  parallel  t(j  Co  c',  etc.,  all  precisely 
as  in  art.  154. 

For  want  of  room,  we  have  reduced  c  -s--^^  s^  ■%  etc.,  by 
one  half. 

Knowing,  in  advance,  that  arches  of  this  kind  are 
inclined  to  the  third  mode  of  rupture,  we  assume  c\  the 
lower  limit,  as  the  point  of  application.  The  resulting 
curve  of  pressure  justifies  the  assumption. 

It  nearly  touches  the  outer  limit  midway  between  the 
key  and  the  spring,  while  at  these  points  it  coincides  with 
the  lower  limit.  Comparing  the  unloaded  and  the  fully 
loaded  arch,  figs.  53  and  54,  we  see  that  the  curve  of 
pressure  has  made  the  greatest  possible  change  in  its 
place,  consistent  with  the  condition  of  remaining  Avithiu 
the  prescribed  limits.  In  this  case  the  intrados  and  ex- 
trados  are  parts  of  different  circles,  and  it  is  obvious  that 
uo  catenarian  curves  could  furnish  a  more  economical  struc- 
ture. Whatever  curves  be  adopted,  allowance  must  be 
made  for  variations  in  the  curve  of  pressure  correspond- 
ing to  variations  iu  the  load,  bearing  in  mind  all  partial 
removals  for  repairs,  which  may  be  made  at  a  future  day. 

The  thrust  F='2a  X  /=2  X  |  the  half  span  X  •^5 /5=270G, 
taken  from  the  drawing,  is  al)out  four  per  cent,  more  than 
the  thrust  computed  from  art.  IIG,  r  =  55^  7^=1.0208  = 

l-f-.     This,  if  we  suppose  each  cubic  foot  to  weigh  ITO 

pounds,  gives  a  mean   pressure  at  the  key,   of  115,000 
pounds  per  square  foot  =  1,000  jtounds  per  square  inch. 


GEOMETlllCAL     METHOD.  419 

and  a  pressure  of  3,200  pounds  per  scpiare  inch  at  tlie 
edge  of  the  key  most  exposed. 

This  is  more  llian  twice  as  great  as  any  pressure  given 
in  table  I,  l)ut  is  little  over  one  tenth  the  strength  of  the 
material  as  determined  hy  Capt.  Meigs.  The  pressure  at 
the  spring,  I'l=2((Xc'  f^,  gives  a  somewhat  larger  pressure 
per  unit  of  surface. 

Were  this  arch,  after  the  removal  of  the  center,  loaded 
progressively  in  horizontal  layei'S,  it  would,  at  one  stage, 
be  surcharged  horizontally  up  to  the  level  of  the  top  of 
the  key,  or  nearly  to  that  level.  At  that  stage  it  would 
be  far  fi'om  practicable  and  barely  possible  (arts.  143, 145). 
It  would  undoubtedly  fall.  Hence  the  necessity  of  put- 
ting on  the  load  in  due  proportion  at  the  reins  and  at 
the  key  simultaneously. 

Ordinary  prudence  would  require  a  larger  arch,  and 
in  fact  the  plan  of  Capt.  Meigs  providi  s  for  a  larger  arch 
of  rubble  masoni-y  resting  on  the  cut  stone  of  the  inner 
arch. 

159.  We  have  supposed  the  curves,  which  mark  the 
nearest  approacli  of  the  curve  of  pressure  to  the  outlines 
of  the  arch,  to  divide  the  joints  into  three  equal  parts. 

The  geometrical  method  does  not,  however,  in  any 
measure  depend  upon  these  proportions.  We  can  adopt 
any  other  proportions;  for  example,  lay  olf  the  limit 
curves  each  at  one  tenth  the  thickness  of  the  joint  from  the 
central  point,  leaving  the  exterior  parts  each  two  fifths  the 
joint. 


RUPTURE    OF    jrASONRY    BY    C0:MPRESSI0X. 

160.  Let  the  vertical  (( J>  I'ising  from  a  solid  foundation, 
and  the  indefinite  horizontal  ///,  l)e  the  outline  of  a  ].iece 
of  masonry  supporting  a  weight  oi-  pressure  distrilnited 


•4-20 


THEORY     OF    THE    ARCH. 


uniformly  and  indefinitely 
along  the  upper  snrfiice  hf. 

The  width  measured  at 
right  angles  to  the  i)laue  of 
the  sectiou:=one  unit. 

^.)=the  pressure  per  unit 
of  surface  on  hf. 

/<t=the  heii^ht  a  h. 

-S'=  the  surface  ahii  cor- 
responding to  any  line  of 
rupture  a  n. 

^y=the  weight  of  each 
unit  square  in  S.  ^^°-  ^^• 

r=the  anHe  h  a  n.  /=fnction=cot.  a. 

a  —  t\\Q  angle  h  af  between  the  vertical  and  the  natural 
sl()pe=90°— the  angle  of  friction  ;  _f=cot.r^==  the  ratio  of 
friction  to  pressure. 

//=the  coherence  of  the  material,  per  unit  of  surface, 
or  the  force  which,  acting  along  any  line  (i  n,  shall  tear 
asunder  one  unit  in  length. 

]r=j)X/t  Xtang.  i'  =  the    whole    weight    or    pressure 
on  h  n. 

P=the  hoi'izontal  force  which,  assisted  hy  friction 
and  Coherence,  shall  hold  in  equilibrium  the  weights  Sx/f 
and  ir  corresponding  to  any  prism  a  h  n. 

We  have,  parallel  to  «  >?,  the  components  Pxsin.  ■i', 

TT^X  COS.  V,  >S'X^>'  X  cos.  i\  a  X  — —  ;  and,  pei-pendicular  to 

cos.  V 

a  n,  Pxcos.  r\  irxsin.  i\  ^S'x;/Xsin.  v.     For  equilibrium 
we  have  P  sin.  /--j-y  x  — '  -  +  (P    cos.  r+  W  sin.  r-\-Sx 

COS.  /' 

^/Xsin.  r)  /=  W  coii.T-\-Sxp'  COS.  ?•,  which,  substituting 
i'>r  ir,^>Axtang.  v  ;  and  for  A',  j;Jr  tang,  v  gives 
r=2)^i  tang,  t' X  tang.  {a  —  r^-{-^pli^  tang.  ^'X 


tang,  (a-c)- 


(jJi  sin.  a 
COS.  yXcos.  (a—v)' 


(''■•) 


GEOMETRICAL    Min'IIOD.  421 

We  readily  find,  by  tlie  calculus,  or  ti'igonometry,  or 
plane  geometry,  that  tang.  ?'Xt:uig.  (a— t')  and  cos.  vjX 
COS.  («— f)  are  both  maxima  when  v=j^a.  Hence,  the 
greatest  horizontal  force  which  the  weights  W  and  Sxp' 
can  cause,  is 

F=ph  tang.^  \a-{-\2^'h^  tang.^  ^a  —  ^hjli  tang.  \<(.     (77) 

If  j9  or  TF  be  nothing,  we  have  the  thrust  of  an  em- 
bankment of  earth  ;  and  we  see  that  the  angle  of  greatest 
thrust  is  not  affected  Ijy  any  supposed  cohesion  in  the 
mass. 

If  W  be  so  large  that  S  in  comparison  may  be 
neglected,  we  have  the  thrust  of  a  column. 

P:=p]i  tang.-  la  —  2(jh  tai]g.  Ul  (7S) 

Let  l=t\ie  width  of  the  column = A  tang.  i(/.  AVe 
have 

F=  TFxtaug.  \a-1(jl  (79) 

Let  us  suppose  the  weight  or  pressure  TFjust  sufficient 
to  overcome  the  tenacity  of  the  material;  we  have 

P=0,  and  TFxtang.^«  =  2^^;  (80) 

giving  the  weight  that  may  be  supported  when  the  ten- 
acity and  the  angle  of  friction  are  known,  and  either  of  the 
latter  when  the  other  and  the  weight  are  known. 

The  angle  of  friction  in  stone,  is  from  30°  to  36°,  giving 
<:^=:G0°  to  54°,  and  the  angle  of  rupture  or  angle  of  least 
resistance  from  30°  to  27°. 

We  are  at  liberty  to  su])pose  the  line  a  h  to  Ijc  the 
surface  of  an  arch— the  intrados  or  the  extrados,  h  n  any 
joint  of  the  arch,  W  the  pressure  at  that  joint ;  and  we 
learn,  from  the  above,  that  the  masonry,  if  too  weak  to 
withstand  the  thrust,  will  give  way,  by  sliding,  in  a  direc- 
tion inclined  about  25°  or  30°  to  the  line  of  the  intrados 
or  extrados. 

We  are  indebted  to  Mosely  for  the  application  of 
this  principle  to  columns,  and  to  experiments  to  determine 


422  TUEORY     OF    THE     ARCH. 

the  resistance  of  mateiials  to  criisliing,  as  it  has  Leen  im- 
properly called.  He  Las,  in  substance,  given  (79);  and  his 
remarks  thereon  have  suifgested  the  whole  of  this  article. 


THE    CURVE    OF   PRESSUKE  "IX    THE    PIER. 

101.  The  equation  of  that  curve,  art.  04,  is 

elt-\-n 

Suppose  li  as  well  as  ./'  varial)le,  ^.constant,  J  the  con- 
stant difference  between  I  and  //,  so  that  l—liiid. 

Develop  (32),  for  x  substitute  x-\-\<i——,  ''^i^l  for  //, 


e 


It'--.    There  results 


a;'7/=^//+— 4 — \-^<l\—a  constant. 
e      t\e    J 

This  is  the  efpiation  of  an  equilateral  hyperbola  re- 
ferred to  a  horizontal  axis  -^d  above  the  point  of  ap[)lica- 

e 

tion  of  the  thrust,  and  to  a  vertical  axis ?—  outside  the 

e 

exterior  face  of  the  pier,  iuside,  should  that  (piantity  be 
neo-ative.  If  F=U-  or  e—\  •T?' the  vertical  asymtote 
will  coincide  with  the  exterior  ftice  which  the  curve  ot 
pressure  will  meet  at  an  infinite  distance. 

Vi  F  be  less  than  W  the  asymtote  will  be  within  the 
pier.  Mosely,  in  a  different  manner,  has  obtained  the 
same  result,  a  fact  of  which  we  were  not  aware  in  writing 
the  above. 

102.  The  curve  of  pressure  in  an  arch  very  heavily 
loaded,  is  very  nearly  tlie  common  parabola. 


THE     CIRCULAR     KING.  403 

The  most  economical  curves  of  the  intrados  and  extra- 
dos  would,  in  this  case,  also  he  })ai'abo]as. 

At  the  other  extreme,  an  unloaded  arch  iniinitcly  ihin, 
the  curve  of  "equilibrium  is  the  catenary. 

16 


■424 


THEORY    OF    THE    ARCH. 


ACTUAL  THRUSTS. 
Table  AA. 

Table  of  TJirust-s  of  the   Unloaded  Ci radar  Ring. 

Tlie  first  column  gives  the  ratio  of  i?,  the  radius  of 
tlie  extrados,  to  ;•,  the  radius  of  the  intrados. 

The  second  cohimn  gives  the  ratio  of  the  diameter, 
2/',  to  tlie  thickness  at  the  key. 

The  third  column  gives  the  new  thrust  based  on  the 
condition  that  no  joint  shall  open  ;  the  curve  of  pressure 
approaching  the  extrados  at  the  key,  within  one  third 
the  length  of  the  joint,  and  the  intrados  at  the  reins 
within  one  third  the  length  of  the  joint ;  this  thrust=^. 

The  fourth  column  gives  the  thrusts  of  table  x\,  for 
the  same  values  of  A"^,  calculated  on  the  supposition  of 
actual  rupture,  the  curve  of  pressure  passing  through  the 
extrados  at  the  key  and  the  intrados  at  the  reins ;  this 
thrust  =ri^'. 

The  fifth  column  gives  the  ratio  ^=  7^„,  of  these  two 
thrusts. 

The  values  of  F  and  h  are  a  little  in  excess,  tlie  ex- 
cess increasing  with  K. 


"< 

^,^ 

Values  of  Jf' 

-0 

t. 

Values  of/"' 

*o 

II 

•        1 

Values  of  F. 

from 

II 

K 

^   1^ 

Values  of  J''. 

from 

II     t 

^ !  V 

"5  .1 

Table  A. 

fe,!^ 

<N   ft; 

Table  A. 

m^. 

(1) 

^ 

(3) 

(4) 

(5) 
1.065 

0) 

(2) 

(3) 

(4) 

(5) 

1.01 

200. Of 

r'x  0.00941; 

r'x  0.00889 

1.21 

9.52 

r'x  0.1 6437 

r''x0. 11516 

1 .  427 

1.02 

lOO.OC 

0.01 HlH 

o.ditjyi 

1.075 

1.22 

9.09 

0.17151 

0.11887 

1.443 

1.03 

66. 6« 

0.026y3 

0.02459 

1.095 

,1.23;8.69 

0.178(50 

1.464 

|1.04 

50.00 

0.03504 

0.03139 

1.116 

1.24  8.33 

0.18555 

&c.,  ns  in 

1.48l| 

1.05 

40.00 

0.04 34s 

0.03813 

1 .  140 

1.25  8.00 

0.19255 

table  A. 

1.499 

1.06 

33.33 

0.O5178 

0.04455 

1.162 

ll.267.69 

0.2OO1-J 

1.521 

il.07 

28.57 

0.0.')98H 

0.05065 

1.18-J 

1.27  7.40 

0.20624 

Anirlos  of  rup- 

1.536 

(1.08 

25.0(r 

O.0t;8(i4 

0.05649 

1.20-1 

1.2817.14 

0.212H1 

ture  the 

1.554 

1.09 

22.2'.i 

f».<i7r.n7 

(1.06177 

1.23-j 

1.29  6.89 

0.21925 

^anlo  ns  in 

1  .  565 

,1.10 

20.  Oc 

0.(»H3f.<t 

0.06754 

1 .  239 

1.3o|6.66 

0 . 22632 

table  A. 

1.579 

tl.ll 

18.18 

O.O'JlOo 

0.07273  1.252 

1.3l;6.45 

0.23338 

1.608 

1.12 

16.6ti 

0.oHH4/i 

0.07789  1.264 

1.32,6.26 

0.24050 

1.631- 

1.18 

16.3^ 

O.Ki.'jHT 

0.082.')4 

1.28« 

1.33,6.06 

0 . 24965 

1.676 

1.14 

14. 28 

0 .  n  33o 

0.0872'.* 

1  .  298 

1.345.88 

0.25501 

1.685 

1.15 

13.33 

0.12n7<; 

0.09176 

1.316 

|l. 3515. 71 

0.2623C. 

1.7U: 

l.Di 

12. 5t 

0.12821". 

0.09593 

1.337 

1.36  5.55 

0.267 SI 

1.73f 

1.17 

11.7(1 

{)  .\'.\r,Ti',\ 

'1.352 

1.37  5.40 

0.2753:! 

1.757 

1.18 

11.11 

0.14281 

4c..  ns  In      \\  .37  1 

1.38  5.26 

0.2828(1 

1.785 

;i.i9 

10.53 

0.1.^024 

table  A.      1 1  _  3y.j 

1.39  5.13 

0.28849 

1.801 

1.20 

10. 0( 

0.15727 

il.4)2 

1.40!5.00 

0.29616 

1.8S2 

TABLE     DD. 


425 


Table  DD. 

Table  of  the  actital  tlira-^ts  of  semicirciddr  arclies  •s-ur- 
cliarged  1  lor i?:on tally ^  tlie  curve  of  i^ressure  pas-mig 
tlirougli  the  middle  of  the  hey  and  the  middle  of  the 
loeahest  joint  (see  art.  IIG,  an  explanation  of  the 
columns). 


Ratio  of 

0  o'o- 

Value.s  of 

the  span 

e^f 

Values  of  F.^ 

T     F 

Values  of 

A.J  from  table 

of 

to  the 

v.zl 

Values  of  F= 

from  table  I)  = 

''=F. 

^l  =  tlie  maxi- 

7*; =  the 

Values 

thick- 

s«5- 

the  maxinmui 

the  maximum 

■^   2 

mum  effect 

maximum 

of 

K= 

ness  at 

"  1  2 

thrust  down 

thrust  in  the 

or 
coeffi- 
cient of 
stabiry. 

of  a  surcharge 

effect  of  the 

''< 

1+^ 
r 

the  key 
2r 
~  d 

a)  -fi  & 

to  «  =  00°. 

case  of  rupture 
and  fall. 

of  constant 
depth  t. 

siirchar<;e  in 
tlie  ca.se  of 
rupture  and 

C  j;  o 

73° 

fall. 

1.00 

?--x  0.05563 

r-x  0.05547 

1  . 0(  t 

rtx  1.0000 

1.00 

1.01 

200.00 

71 

0.06315 

0.06132 

1.03 

1.0050 

1.16 

1.02 

100.00 

70 

0.07083 

0 . 06647 

1.06 

1 .0099 

ri!x  0.8187 

1.23 

1.03 

66.67 

69 

0.07865 

0.0718.5 

1.09 

1.0148 

1.30 

1.04 

50.00 

67 

0.08668 

0.07680 

1.13 

1.O190 

0.7531 

1.35 

1.05 

40.00 

66 

0.09484 

0.08175 

1.10 

1.0244 

1.40 

1.06 

33.33 

65 

0.10317 

0.08638 

1.19 

1.0291 

0.7059 

1.46 

1.07 

28.57 

63 

0.11165 

1.22 

1.0338 

1.50 

1.08 

25.0.0 

62 

0.12029 

Ac,  from  4th 

1.27 

1.0885 

0.6678 

1 .55 

1.09 

22 .  22 

61 

0.12908 

column  of 
table  D. 

1 .  30 

1.0431 

1.60 

l.K 

20.00 

59 

0.13801 

1.34 

1.0470 

0 . 6353 

1.65 

1.11 

18.18 

58 

0.14709 

1.38 

1.0521 

0.6206 

1.70 

1.12 

16.67 

57 

0.15633 

1.43 

1.0506 

0.6O68 

1.74 

1.13 

15.38 

56 

0.16571 

1.47 

1.061t» 

0.5930 

1.79 

1.14 

14.28 

55 

0.17522 

1.51 

1.0054 

0.5810 

1.83 

1.15 

13.33 

53 

0. 18490 

1.56 

1.0698 

0.5690 

r.88 

1.16 

12.50 

52 

0.19471 

1 .  60 

1.0741 

0.5575 

1.93 

1.17 

11.76 

51 

0 . 20468 

1.65 

1.0783 

1.97 

1.18 

11.11 

50 

0.21479 

1.70 

1.0826 

2.02 

1.19 

10.53 

49 

0.2*2504 

1.75 

1 . 0808 

&c.,  from  the 

2.07 

1.20 

10.00 

47 

0.23544 

1.80 

1.090'.> 

VMh  column 
of  table  F. 

2.12  ' 

1.21 

9.52 

46 

0.24598 

1.80 

l.oi).5o 

2.17   1 

1.22 

9.09 

45 

0.25667 

1.91 

1.0991 

2.22  ! 

1.23 

8.69 

60 

0.25960 

1.91 

1.1081 

2 .  27 

1.24 

8.33 

60 

0.26935 

1.90 

1.1071 

2.32 

1.25 

8.00 

60 

0.27916 

ti.Ol 

1.1111 

2.37 

1.26 

7.69 

60 

0.28900 

2.07 

1.1150 

1.27 

7.40 

60 

0.29893 

2.12 

1.1189 

1.28 

7.14 

60 

0.3U889 

2.18 

1.1228 

1.29 

6.89 

60 

0.31892 

2 .  24 

1.1260 

1.8( 

6.67 

60 

0.32899 

2 .  30 

1.1304 

1.31 

6.45 

60 

0.33911 

2.36 

1.1342 

1.32 

6.26 

60 

0.. 34929 

2.42 

1.1379 

1.33 

6.06 

60 

0.35952 

2.49 

1.1416 

1.34 

5.88 

60 

0.36980 

2.55 

1.1453 

1.35 

5.71 

60 

0.38013 

r^'x  0.14666 

2.59 

1.36 

5.55 

60 

0.39051 

0.15111 

2.58 

1.37 

5.40 

60 

0.40095 

i-c,  from  5th 
column  of 

2.58 

1.38 

5.26 

60 

0.41143 

2.57 

1.39 

5.13 

60 

0.42196 

table  D. 

2.56 

1.40 

5.00 

60 

0.48254 

2.56 

426 


TUEORY     OF    THE     ARCH. 


Table  FF.     {See  art.  117.) 

The  ]\rAGAzixE  or  Koof-covkred  Circular  Arch  of 
180°,  WITH  A  load  of  masonry,  or  of  equal  weight 

AVITH    .MASONRY,  RISING,  ON    EACH    SIDE    OF   THE  CENTRAL 
KIDGP:,    TO    A    KOOF   TANGENT    TO    THE    EXTRADOS. 

Actual  thn/6-t  ill  four  -s•?/.s•/e>/^<?,  tlie  curve  of  pressure  ixi-'ising 
at  one  third  the  length  erf  the  joint  from  the  extraJo-s  at 
the  leey^  and  from  the  intrados  at  the  joint  of  greatest 
thrust. 

[/=the  angle  between  the  roof  and  a  vertical ;  r=tlie 
radius  of  tlie  intrados;  7?=t]ie    i-adiiis   of  tlie  extrados  ; 

(/=tlie  tliiekuess  of  the  arch  proper  at  tlie  key;  /r=_^= 

1+-;   C'=the  decimal;  i"=the  thrusts/- X  6';  .■l  =  the 

addition  to  the  thrust  caused  hy  a  snrchai'ge  of  the  con- 
stant depth /f  above  the  tangent  roof;  C=  the  decimal  in 
the  last  column  ;  A=^rtC,'  ^=the  coefficient  of  stability  = 
the  quotient  of  the  thi'ust  in  this  table,  divided  by  the 
thi-ust  of  the  same  arch  in  table  F=the  ratio  of  the  actual 
thrust,  on  the  conditions  expressed  above,  to  the  theoretic 
thrust  at  the  instant  of  rupture  and  tall.  The  angle  of 
maximum  thiust  is  measured  from  a  vertical.] 


7=60'" 

/=55* 

1=^0'          1          1=4:,' 

Siir< 

hargc. 

1         r 
1.1 

11 

F= 
thrii.st  = 

6 

c  C 

"si  . 

F^ 
thrii8t= 

i 

o  — 
Ml  . 

3  >'■ 
£ 

F= 

thnist= 

S 

2  tZ 
<  a 

E 

F= 

thru.''t  = 

6 

5 

^=    J 
effect  of 

surcb'ge 

=  rW. 

C 

C 

C 

C 

C 

1.40 

.50^ 

"t.:i.54'.t  1 

841.50° 

0.396111.81 

47  r 

0.4523 

1.78 

47+' 

0.5273 

1.76 

52" 

0.6622 

1.39 

" 

•  ».;J4(S2;1 

.82    •• 

0.3892  1.78 

" 

0.4452 

1.76 

0.5196 

1.75 

" 

0.6654 

.   1.38 

" 

0.3415  1 

.79 

" 

0.3823  1.76 

" 

0.4381 

1.74 

" 

0.5119 

1.72 

51 

0.6687 

1.37 

" 

0.3347  1 

.77 

" 

0.3754  1.74 

" 

0.430« 

1.72 

45 

0 . 5042 

1.71 

" 

0.672(1 

,   1.36 

47+° 

0.3281 ;i 

.75 

" 

0.3685  1.73 

45 

0.4234 

1.71 

" 

0.4968 

1.69 

" 

0.6753 

,   1.35 

0.3214 

.73 

17+ 

0.3615  1.71 

" 

0.4159 

1.69 

" 

0.4893 

1.67 

«' 

0.6788 

:   1.34 

" 

0.3147 

.71 

0.3546  1. 69 

<' 

0.4(»K9 

1.67 

" 

0.481S 

1.66 

50 

0.6822 

1.33 

" 

0.307 'J 

.09 

0.3476  1.67 

" 

0.401H 

1  .  65 

" 

0.4742 

1.65 

" 

0.6857 

1.32 

« 

0.3011 

.67 

" 

0.34O6J1.05 

" 

0.3946 

1.64 

" 

0.4666 

1.63 

" 

0.6893 

;   1.31 

" 

0.2943 

.65 

" 

0.3336 11.63 

" 

0.3873 

1.62 

42+ 

0.4590 

1.61 

49 

0.6929 

TABLE    FF.— CONTINUED 

4. 

27 

1                 1 

/=fii)' 

7=55' 

/=50- 

I=Ah'             j    Sure 

liarfre.    1 ) 

<\      of 

a 

F= 

hrust= 

6 

^1 

1'= 

thrusts 

S 

0  E 

5  '''■ 

"  2 

F= 

hrust=: 
r-  C. 

& 

s 

F= 

hru8t= 

i 

A=     ii 
eflect  of 
surch't'i'  1 

i 
1 

r 

C 

0 

C 

0 

^i 

"'      1 

1.30 

47*° 

0.2874 

1.63 

45° 

0.3266 

1.61 

45° 

0.3798 

1.60 

42+° 

).4617 

1 .  60 

49° 

0.6966' 

1    1.29 

45 

).2805 

1.61 

" 

0.3196 

1.6(1 

42+ 

).3729 

1.59 

" 

).4443 

1.59 

" 

0.7004 

1.28 

" 

0.2737 

1.59 

" 

0.3126 

1.5-8 

).3658 

1.58 

" 

0.4309 

1.57 

48 

0.7042j 

1.27 

" 

0.2669 

1.57 

" 

0 . 3056 

1.56 

" 

) . 3586 

1.56 

" 

0.4294 

1.56 

" 

0.708'Ji 

1.26 

" 

0.2599 

1.55 

" 

0.2986 

1.55 

" 

i».3514 

1.55 

40 

0.4219 

1.54 

" 

0.7122' 

1.25 

42* 

0 . 2529 

1.53 

42  ^ 

0.2916 

1.53 

" 

0.344(1 

1.63 

" 

0.4148 

1.53 

47 

0.7163i 

1.24 

" 

0.2461 

1.52 

0.2846 

1.52 

40 

0.3372 

1.62 

" 

0.4076 

1.52 

" 

0.72061 

1.23 

" 

0.239211.6(1 

" 

0.2776 

1.50 

" 

0 . 3303 

1.60 

" 

0.4003 

1.50 

« 

0.7249| 

1.22 

" 

0.2323 

1.48 

" 

0.2706 

1.49 

" 

0.3233 

1.49 

" 

0.3930 

1.49 

46 

0.729R| 

1.21 

«' 

0.2253 

1.47 

" 

0.2636 

1.47 

374 

0.3161 

1.48 

37+ 

0.3860 

1.48 

" 

0.733SI 

1.20 

40 

0.2184 

1.45 

40 

0.2566 

1.46 

" 

0.3089 

1.46 

" 

0.3790 

1.47 

45 

0.7387J 

1.19 

" 

0.2115 

1.44 

'< 

0.2498 

1.45 

" 

0 . 3023 

1.45 

" 

0.3720 

1.46 

0.7435 

1.18 

" 

0.2045 

1.42 

" 

0.2430 

1.43 

35 

0.2956 

1.44 

35 

0 . 3650 

1.44 

44 

0.74861 

1  1.17 

37+ 

0.1976 

1.41 

" 

0.2362 

1.42 

" 

0.2889 

1.43 

" 

0.3584 

1.43 

" 

0.753H' 

1.16 

0.1907 

1.40 

" 

0 . 2294 

1.41 

" 

0.2820 

1.42 

" 

0.3518 

1.42 

43 

0.759-J 

1   1.15 

" 

0.1837 

1.38 

35 

0.2226 

1.4( 

" 

0.2750 

1.41 

324 

0.3451 

1.41 

" 

0.7651 

1.14 

35 

0.1770 

1.37 

" 

0.2164 

324 

0.2690 

" 

0.3389 

1.40 

42 

0.771L 

1.13 

32+ 

0.1701 

1.36 

" 

0.210( 

" 

0.2629 

" 

0.3327 

1.39 

41 

0.777.V 

1.12 

« 

0.1635 

1.35 

" 

0.2036 

30 

0 .2567 

30 

0 . 3266 

1.38 

" 

0.7841j 

1.11 

30 

0.156& 

1.34 

" 

0.1971 

" 

0 . 2505 

" 

0.3211 

1.37 

40 

0.7912i 

1.10 

" 

0.1 5041 1.33 

30 

0.190-4 

" 

0.2441 

*' 

0.3155 

1.35 

39 

0.798h 

428 


THEORY     OF    THE    ARCH. 


Table  DDD. 

TaUe  of  the  actual  thrusts  of  set/i  icircular  arches  surcharejed 

horizontally^  the  curve  of  j^f'^ss-uj-e  jya-ssinff  at  J  the 

length  of  the  joint  from  the  extraJos  at  the  Tcey^  and 

from  the  intrados  at  the  joint  of  <jreattst  thrust^  cOc, 

<£'<?.     See  the  explanation  in  art,  118. 


F^r-^C. 
C. 

A=rtC. 
C. 

Angle  of  maximum  thrnst 

<?,or  coeft  of  stability. 

Values  of 

r 

In  the  case 
of  actual 
ruiiture 
and  full. 

Table  D. 

The 

curve  of 

pressure  at 

one  third 

the  lenftth, 

Ac., 
as  above. 

Cnr>e  of 
pressure 

at  the 
middle  of 
the  joints, 

Ac, 
Table  DD. 

Curve  of 
pressure  at 

one  tliird 
the  length 

of  the 
joints.  Ac. 

as  above. 

Curve  of 
pressure 
at  the 
middle  of 
the  joints, 

Ac, 
Table  DD. 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

{-) 

(S) 

;     1.00 

0.0556 

1.0000 

75° 

72°  30' 

73° 

1.000 

1.00 

1.01 

0.0625 

0.9248 

74 

" 

71 

1.020 

1.03 

1.02 

0.0694 

0.8971 

73 

70° 

70 

1.040 

1.06 

1.03 

0.0763 

0.8772 

71 

67°  30' 

69 

1.060 

1.09 

1.04 

0.0832 

0.8612 

70 

" 

67 

1.080 

1.13 

1.05 

0.0903 

0.8476 

69 

" 

66 

1.100 

1.16 

1.06 

0.0973 

0.8358 

68 

65° 

65 

1.126 

1.19 

1.07 

0.1042 

0.8252 

67 

" 

63 

1.140 

1.22 

1.08 

0.1112 

0.8156 

66 

" 

62 

1.170 

1.27 

1.09 

0.1181 

0.8069 

66 

62°  30' 

61 

1.190 

1.30 

1.10 

0.1250 

0.7988 

65 

" 

69 

1.220 

1.34 

1.11 

0.1319 

0.7912 

65 

" 

58 

1.240 

1.38 

1.12 

0.1387 

0.7841 

64 

60° 

57 

1.260 

1.43 

1.13 

0.14.55 

0.7775 

64 

" 

56 

1.290 

1.47 

l.U 

0.1523 

0.7712 

64 

" 

55 

1.310 

1.61 

1.15 

0.1590 

0.7651 

64 

" 

53 

1.340 

1.56 

1.16 

0.1657 

0.7594 

64 

" 

52 

1.360 

1.60 

1.17 

0.1723 

0.7539 

64 

" 

51 

1.390 

1.65 

1.18 

0.1788 

0.7486 

63 

57°  30' 

50 

1.410 

1.70 

1.19 

0.1853 

0.7435 

63 

" 

49 

1.440 

1.75 

1.20 

0.1918 

0.7387 

63 

" 

47 

1.470 

1.80 

1.21 

0.1982 

0.7339 

63 

■< 

46 

1.490 

1.86 

1.22 

0.2046 

0.7293 

63 

" 

45 

1.520 

1.91 

1.23 

0.21 09 

0.7249 

63 

" 

1.550 

1.91 

1.24 

0.2171 

0.7205 

62 

1.580 

1.96 

1.25 

0.2233 

0.7163 

62 

" 

1.610 

2.01 

1.26 

0.2294 

0.7122 

62 

" 

1.640 

2.07 

1.27 

0.2355 

0.7082 

62 

« 

1.670 

2.12 

1.28 

0.2415 

0.7042 

62 

" 

1.700 

2.18 

1.29 

0.2475 

0.7004 

61 

" 

1.730 

2.24 

1.30 

0.2534 

0.6966 

61 

" 

1.770 

2 .  30 

1.31 

0.2593 

0.6929 

61 

" 

1.800 

2.36 

1.32 

0.2651 

0.6893 

61 

" 

1.830 

2.42 

1.33 

0.2709 

0.6857 

61 

" 

1.870 

2.49 

1.34 

0.2766 

0.6822. 

60 

It 

1.910 

2.55 

1.35 

0.2823 

0.6788 

60 

" 

1.920 

2.69 

1.86 

0.2879 

0.675.S 

60 

" 

1.900 

2.58 

1.37 

0.2935 

0.6720 

60 

" 

1.880 

2.68 

1.38 

0.2990 

0.6687 

59 

" 

1.870 

2.57 

1.39 

0.3045 

0.6654 

69 

" 

1.860 

2.66 

1.40 

0.3099 

0.6622 

59 

" 

1.830 

2.56 

430 


THEORV     OF    THE    ARCH. 


Table  EE. 
Segmental  Arches  Loaded   up   to   the  Level  of  the  top 
curve    oe    pressure    passing    within    one    third    the 
and    from    the    intrados    at   the    springing    line    or 

[(7=tlie   decimal   in   any  column ;    r=:the  radius  of   tlie 
caused  l)y  a  surcharge  of  the  constant  deptli  t  a1»ove  tlie  key  ; 

1-| — ■  j'  d=the  thickness  at  the  key.     See  art.  120. 


1 

«=4/;  r=2-i-/,- 

8=5/;  r=3p; 

«=6/,-  r=5/; 

8=V;  r=6|./; 

r 

v=5S°  V  80". 

v=48'  86'  lO". 

T=S6°  52'  10'. 

v=SV  68'  27". 

F=r^ 

xC. 

J=HC. 

a 

Jf=r»xC. 

A=rtC. 
C. 

i^=r»x6*. 

A=rtC. 

a 

F=r^  X  C. 

A=rtC. 
C. 

a 

6 

S 

C. 

J 

C. 

6 

1.40  0.3001 

1.827 

0.6624  0.2938  1.736 

0.6470 

0.2700 

1.596 

0 . 6067 

0.244111.443  0.5563 

1 1.39  0.3036 

1.845 

i).6654  0.2887  1.754 

0.6511 

0.2655 

1.613 

0.6115 

0.240311.460  0.5615 

|l.38!0.298-.i 

1.862 

0.6687  0.2836  1.771 

0.6552 

o.261o! 1.630 

0.6163 

(t. 2364  1.477  0.5668 

ll.37|O.'2920 

1.880 

0.6720 

0.27841. 789 

0 . 6593 

0. 2564 1 1.648 

0.6212 

0. 23251 1.494JO.  5723 

11.36  0.2871 

1.900 

0.6753 

0.2732  1.808 

0.6635 

0.2518 

1.666 

0.6262 

0.228511.512(1.5778 

jl. 35  0.2814 

1.918 

0.6788 

0.2679 1.826 

0.6677 

0.2471 

1.684 

0.6313 

0.2244| 1.530  0.5835 

1.34  0.2758 

1.938 

0.6822 

0.2626  1.846 

0.672O 

0.2424 

1.705 

0.6365 

0.220311.548  0.5893 

1.33  0.2700 

1.909 

0.6857 

0.25721.865 

0.6764 

0.2376 

1.723 

0.6418 

0.2162:1.568 

0.5952 

1.32'0.2643 

1.876 

0.6893 

0.2518'l.886 

O.68o7 

0.2327  1.743 

0.6471 

0.2120  1.588'o.6012| 

il.3l|0.2584 

1.842 

0.6929 

0.2463  1.906 

o.6852lo. 227811. 763 

0.6526 

0.2078 

1.6080.6074 

1.30  0.252G 

1 .  809 

0.6966 

0.2407  1.926]o.6897 

0.2229 

1.783 

0.6581 

0.2035 

1.628(».6138 

'1.29  0.2460 

1.775 

0;7004 

0.2351  1.89310.6942 

0.2178 

1.804 

0.6638 

0.1991 

1.65(i[0.6202 

1.28  0.2407 

1.745 

0.7042 

0.2294  1.859 

0.6988 

0.2127 

1.826 

0 . 6695 

0.1946 

1.670  (». 6269 

1.27  0.2346 

1.714 

0.7082 

0.2237  1.826 

0.7035 

i>.2076 

1.849 

0.6754 

0. 190111. 693'o. 6337 

1.26  0.2285 

1.683 

0.7122 

0.2179 1.793 

0.7082 

0.2023 

1.870 

0.6814 

0.1856|1. 715:0. 6406 

1.25j0.2224 

1.654 

0.7163 

0.2120  1.761 

0.7130 

0.1970 

1.883 

0.6875 

0. 1808  1.737 

0.6478 

1.24  0.2162 

1.623 

0.7205 

0.2061  1.730 

0.7179 

0.1916 

1.85(t 

0.6937 

0.1761  1.761 

0.6551 

1.23,0.2099 

1.594 

0.7249 

0.2001  1.700 

0.7228 

0.1862 

1.817 

o.70(»l 

0.1712  1.785 

0.6627 

1.22  0.2036 

1 .  566 

0.7293 

0.1940 1.668 

0.7278 

0.1806 

1.781 

0.7066 

0.1663  I.8I0 

0.6704 

1.21  0.1972 

1.538 

0.7339 

0.1878  1.636 

0.7329 

0.1750 

1.74N 

0.7132 

0.1613 

1 .  835 

0.6784 

1.20'0.1907 

1.510 

0.7387 

0.1816  1.606 

0.7380 

0.1692 

1.713 

0.72O(i 

0.1561 

1.830 

0.6865 

1.19  0.1842 

1.482 

0.7435 

0.1753  1.575 

0.7432 

0.1634 

1.681. 

0.7269 

0.1509 

1.794 

O.6950 

1.180.1776 

1.456 

0.7486 

0.1689  1.546 

0.7485 

0.1575 

1.646 

0.7340 

0.1456 

1.756 

O.7036 

1.170.1710 

1.429 

0.7539 

0.1624  1.515 

0.7539 

0.1514 

1.611 

0.7413 

0.1401 

1.7191 

0.7125 

1.16,0.1643 

1.403 

0.7594 

0.1558  1.485 

0.7594 

0.1453 

1.578 

0.7487 

0.1346 

1.682i 

).7218 

1.15  0.1575 

1.377 

0.7651 

0.1492  1.45C.lo.7651 

0.1391 

1.544 

0.7564 

0.1289 

1.644 

).7313 

1.14;0.1506 

1.351  0.7712 

0.1425  1.425 

IL7712 

0.1327 

1.510 

0.7642 

0.12311 

1.607 

).7411 

1.13,0.1437 

1.326  0.7775 

0.1356  1.397 

0.7775 

0.1263 

1.477 

0.7722 

0.11711 

1.568| 

(.7512 

1.12  0.1367 

1.30]  0.7841 

0.1287  1.36K 

0.7841 

0.1197 

1.442 

0.78O4 

o.lllo 

1.529 

(.7617 

1.110.1296 

1.276  0.7912 

0.1216  1.338 

0.7912 

0.1129 

1.408 

o.788^ 

0.1047 

1.489 

).7726 

1.10  0.1225 

1.250  0.7988 

0.1145 1.310 

0.798H 

0.1061 

1.374 

0.7975 

0.09831 

1.450 

).7839 

1.09  0.1152 

1.226().8o69 

0.1072  1 .279 

o.8ot'.9 

0.0991 

1 .  339 

0.8064 

0.0917 

1.40H 

1.7956 

1.08  0.1079 

1.199  0.8156 

0.0998 1.251 

0.8156 

1.0919 

1 .  304 

0.8156 

0.0849I 

1 .  367 

(.8077 

1.07  0.1005 

1.177,0.8252' 

0.0923  1  .221 

0.8252 

).o846 

1 .  268 

0.8252 

0.0779 

1.325 

1.8203 

1 .06  O.0'.t3o 

1.151  0.83.'-.K 

ti.oH47  1.191  0.835R 

).o771 

1 .  232 

0.835K 

0.0707 

1 .  283 

1.8334 

1.05  0.0855 

1.128  0.8476 

).o77o  1 .163 

0.8476 

).0695 

1.196 

0.8476 

0.0633 

1.239 

).847o 

1.04  0.0778 

1.1030.8612 

1.0691  1.133 

0.8612! 

0.0616 

1.160 

0.8612 

t.0556 

1.196, 

).8612 

1.03  0.07 00 

1.077,0.8772 

0.0611  1.103 

0.8772 

0.0536, 

1.124 

0.8772 

).0477' 

1.150! 

').8772 

1.02  0.0622 

1.0.54  0.8971 

1.0529  l.(»73 

0.8971 

1.0453 

1.084 

0 . 897 1 

1.039^' 

1.103 

>.8971 

1.01  0.0542 

1.027  0.9248. 

>.o446  1.042 

0.9248 

).0368 

1.042 

i).924« 

).0310. 

1.055 

).9248 

1.00  0.0462 

ii.ooool 

0.(|,S61|              1.00(lO|0.028l| 

1.0(tOo 

).0222| 

1 

I. 0000 

TABLE     EE. 


431 


Table  EE. 

OF  THE  KEY.  AcTUAL  TIIKUST  IN"  SEVEN  SYSTEJLS  :_  THE 
LENCtTII  of  the  joint  from  the  EXTRADOS  at  THE  KEY, 
JOINT    OF    GREATEST    THRUST. 

intrados  ;  i^'^ the  thrust ;  .4=the  addition  to  the  tlinist 
F=rxC;   A  =  rtxC;'  6:=the  span;  /=tlie  rise;  /r= 


1.^ 


8=8/;  r=9-f; 

■v=28°  4'  20". 


F=r^xO. 


a 


40 

39 

38 

37 

36 

1.35 

1.34 

1.33 

1.32 

1.31 

1.30 

1.29 

1.28 

1.27 

1.26 

1.25 
1.24 
1.2f 
1.22 
1.21 
1 .  20 
1.19 
1.18 
1.17 


1.14 
1.13 


1.12 
1.11 
1.10 
1.09 
1.08 
1.07 
1.06 
1.05 
1.04 
1.03 
1.02 
1.01 
1.00 


2188 
2156 
2123 
2090 
2056 
2022 
1988 
1953 
1917 
1881 
1844 
1807 
1769 
1739 
1690 
1649 
1608 
1566 
15 
1479 
1434 
1388 
1341 
1293 
1243 
1192 
1140 
1086 
1030 
0972 
,0913 
.0852 
.0789 
.0723 
.0654 
.0583 
.0509 
.0432 
0351 
.0267 
0178 


293 
310 
326 
343 
361 
378 
398 
416 
,436 
,456 
476 
,497 
,519 
.541 
,562 

.585 
.608 
.633 
.657 
.683 
.707 
.735 
.760 
.788 
.788 
.748 
.706 
.663 
.619 
.575 
.529 
.484 
.440 
.390 
.340 
.287 
.  235 
.180 
.125 
.068 


A=rtC. 


O 


5038 
5092 
5148 
5206 
5264 
5324 
5385 
5448 
5513 
5.579 
5647 
5717 
5789 
5863 
594( 
6019 
6101 
6185 
6271 
6361 
6454 
6550 
6650 
6754 
6861 
6971 
7086 
7207 
,7333 
7464 
7601 
,7744 
,7894 
.8051 
.8216 


8=W;  r=W; 
T=ir  87'  10". 


F-r'^v.  C. 


C. 


1745 
1722 
169fi 
1676 
1652 
1628 
1603 
1578 
1553 
1527 
1501 
1474 
1447 
1418 
1389 


8572 
8765 
8971 
92480 
.OOOOlo 


1360 

1330 

1298 

1267 

1234 

1200 

1166 

1131 

1094 

1055 

1015 
,0974 
.0931 
.0886 
,0839 
,0791 
,074(1 
.0686 
.063( 
.0570 
.0507 
.0440 
.OSes' 
.0291 1 1 
.02(i9  1 
.0121 


1 

1 

1 

1 

1 

1 

1 

1.2 

1 

1 

1 

1 

1 

I 

1 

1 

1 

1 

1 

1 

1 


095 
112 
129 
146 
165 
183 
203 
222 
243 
265 
287 
310 
335 
358 
,388 
,411 
,438 
,465 
,496 
.527 
.558 
.593 
,627 
,663 
,699 
,738 
.77 
,818 
.  805 
.751 
.701 
.644 
.584 
.525 
.462 
.397 
.329 
.256 
.173 
.094 


A=rtC 

0. 


r  =  14'  15'. 


F=r'i  X  C. 


a 


4070 
4123 
4178 
4235 
4294 
4354 
4417 
4481 
4548 
4617 
4688 
4762 
4838 
4918 
5000 
5085 
5173 
5266 
5364 
5465 
5571 
5681 
5797 
5918 
6045 
6180 
6321 
6470 
6627 
6794 
6971 
7160 
7360 
7574 
.7804 
8050 
8315 
8601  lo 
.8911  0 
.9248  0 
ooo(t  0 


1276 
1240 
1204 
1168 
1132 
1097 
1062 
1027 
0993 
0958 
0924 
0891 
0857 
0824 
0791 


077  3 

0762 

0749 

0736 

0723 

0709 

0694 jl 

0679  1 

06631 

064611 

062911 


0611 

0592 

0570 

0547 

0523 

04961 1 

0467 'l 


000 
000 
000 

0(M' 

00(  I 
000 
000 
000 
00(» 
000 
000 
000 
000 
000 
000 

020 

050 

080 

112 

147 

184  0 

222 

264 

310 

357 

410 

469 

534 

597 


A  =  rtC. 
C. 


.0436 
.0400 
.0361 
.0316 
.0265 
.0206 
.0135 
.0051 


668 
749 
830 
930 
938 
843 
,752 
,637 
,606 
,373 
.205 


3538 
3512 
3487 
3462 
3437 
3411 
3386 
3361 
3335 
3310 
3285 
3260 
3234 
3209 
3184 
3159 
3133 

3134 
3220 
3312 
3410 
3516 
3629 
37  Sf) 
3880 
4022 
4176 
4342 
4524 
4725 
4947 
5192 
5467 
5775 
6125 
6524 
6984 
7521 
8155 
8916 
OOOt. 


432 


THEORY     OF    THE     ARCH. 


Elements  of  particular  Bridges — the  dimensions 

[/•:=t]ie  liaU-sjian  in  c'lli])ti(.'al  ai'ohe>i. 


N  AVK  <ilt   ^^l•AII^•^■. 


Date. 


Thickness         Ratio  of  the 


Architect. 


Intrados. 


Rise. 


Feet 


Span. 


Feet. 


at  the 
key. 


Feet 


1  BisliDji  Auckliintl,  over  the 
I     Wear 

2  Llanvrast,  in  Denbighshire . 

:i  \Te.<tniinster  Bridge,  central 
arch 


4  Taaf,  South  "Wales. 


f)  Wcllinirton  Bridge,  overthe 
Aire,  at  Leeds 

C  Waterloo  Bridge,  nine  equal 
arches 


1388 
1636 

1750 
1755 


.  ISegniental . . . 


Inigo  Jones. . , 


1811 


London  Bridge,  central  arch.  1831 

Staines  Biidge,  five  equal| 

,     arches 1832 


Labelye. 
Edwards. 

Rennie. . 
do     .. 


do       ... 

Semicircle . . . 
Segmental . . . 

do       ... 
Scmi-e'lipsc.  . 


Chester  Bridge 1825 

Edinburgh 

Hutcheson  Bridge,  Olasgowl . 

Wliitaddcr    Bridge,   Allan 

ton '. 1842 

Rnihvay  Bridge,  at  Maiden- 

hea.l 

Bridtre  at  Neuilly,  over  the 

Seine,  five  ecjual  nrclics. .  1774 
FJridge   of  IVsnies,   on    thei 

OugQon 1772 


George  Rennie.  do 

Rennie Segmental. . 

Harrison do 

Mylne Semicircle. . 

Robt.  Stevenson  Segmental , . 

Do  &,  Sons do 

Brunei Semi-tllipse . 

Perronet do 


Chateau  Thierrj- 
Louis  XIX 


1786 
1791 


1 H  N'einours 1805 

I'.t  Turin 


Bert  rand.. 


Segmental. 


Perronet Semi-eflipse. 

do       Segmental. . 


do 
Mosca . 


do 
do 


22.00 
17 .00 

38.00 
35.00 

15.00 
35.00 

38.00 
9.25 
42.00 
36.00 
13^ 

11.50 
24.25 
32.00 

17.00 
9.75 

3.75 
18.00 


100y% 

58.00 

76.00 
140.00 

100.00 
120.00 

152.00 
74.00 

200.00 
72.00 
79.00 

75.00 
128.00 
128.00 
44.75 
51.00 
94.00 

63.25 
147.60 


1  10 
1.50 

7.60 
2.50 

3.00 
4.75 

5.00 
3.00 
4.00 
2.75 
3.50 

2.50 

5.25 

5.25 
«io 

4.00 
Z^ 

4? 


Feet 


iliam.  of 
curv.  of  I 

tliu  iii- 
'  tradns 
;  III  the  I 
the  k.  •- 

to  t!, 
tliiek 


14.00 
2.50 

7.00 
8.00 

10.00 
6.0 
6.0 


4.50 


3.0( 


54.77 
38.66 

10.00 
56.00 

33^ 
25.26 

30.40 
•24§ 
50.00 
26.18 
22 .  57 

30.00 


7.1624.38 
.....  24.38 


11.67 

12.75 

.  25.64 


10.81 
31.07 


74.50 
44.30 

10.00 
70.00 

60.55 


52.42 
70.00 
26.18 
37.24 

53.51 


35.00 

64.46 

60.88 
67.49 


ELEMENTS     OF     PARTICULAR    BRIDGES. 


48^ 


COMPILED    MAINLY    I'EOM   CkESY's    ENCYCLOPEDIA. 

/'=:tlie  radius  of  tlie  iutrados  in  segmental  arclies.] 


1 

ai 

Wrt 

a  g 

S  So 

n.s 

Thrust  at  the 

i.fl''^ 

Nature  of 

s-s^ 

'S   C3 

1 

1        key  for 

of  the 

.2  >-  * 

£  B  g 

'    each  foot  of 

?  g" 

material 

~  s:  2 

^-a  S 

REMARKS. 

I      bridge  in 

ca  *-. 

\vlien 

3  ■"  S 

width. 

kuovvn. 

E.S-S 

.2  o  2 

rx  0.06356 

25869 

4000 

11.00 

ProLaljly  Avitli  little  surcharge. 

i 

0.08960 

10560 

4000 

27.00 

"  The  road-way  approached  a  horizontal  line  "  in  con- 
sequence of  the   substitution  of  vehicles  for  pack 
horses. 

'         0.19180 

1 

5908 

4000 

49.00 

Thickness  taken  from  the  drawing  in  Cresy's  Encyclo- 
pedia.    Surcharged  horizontally. 

!         0.07151 

35072 

4000 

8.00 

Fell  on  the  removal  of  the  center,  the  crown  rising ;  but 

stood  after  being  rebuilt  with  hollow  spaces  iu  the    j 

1 

surcharge  over  the  reins. 

0.05764 
0.17113 

25362 

4000 
6000 

20.00 

Surcharged  horizontally. 

Counter  arch  over  the  piers  to  receive  the  horizontal 

21905 

Granite. 

thrust.     Settled  but  a  few  inches  on  removmg  the 

centei". 

0.16723 

3092G 

do 

6000 

14.00 

Settled  at  the  key  only  2"  on*  removing  the  center.  , 
Counter  arches  6  feet  thick  over  the  piers. 

1         0.05721 

18862 

5000 

20.00 

0.06406 

50223 

6000 

c  6 

The  crown    settled  only   2^  inches  on   removing  the 
center.                                                                                   ' 

1          0.11683 
0.07810 

8809 
15177 

4000 
4000 

33.00 
19.00 

1 

1, 

0.06195 
0.25292 

17740 
24666 

400() 
1200 

16.00 

1 

Brick. 

Six  longitudinal  walls  support  the  railway. 

0.19870 

24804 

4000 

11.61 

Settled  2  feet  at  the  crown  on  removing  the  center. 

^         0.06726 

12644 

400(t 

23.00 

Settled  considerably  at  the  crown  on   removing   the  j 
center,  the  abutments  moving  laterally.                          ! 

;         0.26476 

6886 

4000 

42.00 

1 

0.04513 

27493 

4000 

10|- 

In  calculating  the  last  column,  the  pressure,  per  unit 
of  surface,  at  the  extrados  of  the  key,  has  been  as- 

1 

sumed  at  twice  the  mean  i)ressure.                                   1 

0.03877 

18200 

400i 

16.  OC 

0.04887 

42294 

600( 

I'H 

The  weight  of  a  cubic  foot  of  stone  is  assumed  to  be 
160  pounds.     Brick  masonry  125  pounds  per  cubic 
foot. 

INDEX. 


Arcli— liow  distinguished  from  the  beam 
"  "     represented 

Circular  arches  of  180°  without  load- 
thrust,  ultimate,  by  rotation, 

u  "  "  including 

effect  of  mortar,  .         .         •         • 
thrust,ultimate,  by  rotation,  including 

effect  of  surcharge,  . 
thrust,  actual,  by  rotation,      . 

"  "  sliding, 

<i  "  "    including  effect 

of  surcharge 

thrust,    actual,  by   sliding,  including 

effect  of  mortar,  .         .         •         • 
thrust— when  due  to  rotation,  when 

to  sliding,  .         .         •  • 

thrust  by  rotation,  when  zero,    . 
thiclcness  of  pier,    .         .         •         • 
"  "      limit, 

"  arch     "        .         .         • 

geometrical  method,    . 

Circular  arches  of  180°,  surcharged  hori- 
zontally— 
thrust,  ultimate,  by  rotation, 
"        actual,  " 

"  "  by  sliding, 

thickness  of  arch— general  rules,     . 
"  "        limit, 

"  pier,     .         .         .         • 

geometrical  method,    . 

(For  further  particulars,  see  magazine 
arch.) 
Coefficient  of  stability. 
Coulomb — theory  of,        . 
Curve  of  pressure,       .         .         ■         • 

"  "  limit  of  approach  to 

the  extrados  and  intrados,  * 
Curve  of  pressure — equation  of,  in  the 

arch,        .         .  •         •         •   _      • 

Curve  of  pressure — equation  of,  in  the 

pier, 


raragi-aiili  and  Table. 

1,  2. 
1. 


28—35,  38,  A. 

31,  32. 

34,  35. 
114,  AA. 

36—38. 

39. 

3Y,  38. 

41. 

30,  33,  40. 

42 — 45,  B. 

43,  46. 

47. 

153,  154. 


57,  D. 

115,  116,  118. 

DD,  DDD. 

57,  D. 

136,  137,  151,152. 

57,  142,  144. 

64,  65,  12S— 135. 

155. 


17_19,  122— 12G. 

3. 

108—162. 

111. 

147. 
161. 


Page. 


199,  201. 
199. 


216—224,  227. 

219—221. 

222 — 224. 
334,  335. 
225—227. 

227—228. 

226,  227. 

228. 

218,  221,  228. 

229—232. 

230,  233. 

233. 

404. 


243. 
335—343,  346. 

243. 

372—376,  401—404. 
243,  380,  386. 
253—258,  359—371. 

414. 


210,  211,  352—359. 

201. 

324—423. 

328. 

391. 

422. 


436 


INDEX. 


Paragraph  ami  Table. 

Page. 

Curve  of  pressure,  necessary  situation  of, 

ir,-i 

402. 

"                 "         ia  the  arch — liiuits, 

lOii 

422. 

Definitions  and  general  remark*, 

1. 

199. 

Elliptical  arches  without  load — 

ultimate  rotation  and  sliding  thrusts, 

00- 

-04,  A', 

II. 

201- 

-207 

,  204,  323. 

actual  thrusts,      .... 

121 

350. 

tliickness  of  jiier,     .... 

9.5, 

128. 

297, 

359. 

arch, 

130 

138. 

372, 

376. 

Elliptical  arches  with  elevated  ridge — 

ultimate  thrust — rotation,    . 

96, 

97. 

297. 

thickness  of  ])ier,     .... 

98. 

128. 

3oo, 

359. 

geometrical  method,    . 

155 

414. 

Elliptical   arches    surcharged    horizon- 

tally- 

ultimate  rotation  thrust,  etc.. 

09- 

-106,  G 

301- 

-308 

,  322. 

actual  thrust,       .... 

121 

350. 

thickness  of  pier,     .... 

107 

,  128,  133,  135. 

3o8, 

350, 

363,  370. 

"            arch, 

136 

.138,151. 

372. 

376, 

402. 

compared  with  segmental  arches,  . 

100 

—102. 

303, 

3U4. 

geometrical  method,    . 

155 

414. 

Geometrical  method,  of  univer.sal  appli- 

cation,           

153 

—159. 

404- 

-410 

Joint  of  rupture,  rotation,  . 

5,  g 

,  0,  10. 

202, 

203, 

204. 

"               "         sliding. 

2<>. 

212. 

"               "         rotation,  circular  ring. 

28, 

114,  A, 

A.\. 

216, 

334. 

"               "         s  iding,           "           " 

36, 

A. 

225. 

"               "         rotat'n,  magazine  arch, 

48, 

117,  F, 

FF. 

•236, 

344. 

"               "         sliding,           "           " 

52, 

F,  FF. 

230. 

"               "         rotation,  circular  arch 

surcharged  horizontally, 

57, 

D,  DD, 

DDD. 

243. 

.fuint   of  rupture,  sliding,  circular  arch 

surcliarged  horizontally. 

57, 

I). 

243. 

Joint   of  rui)ture,   rotatfon,    segmental 

ring 

60, 

A,  E. 

265. 

Joint  of  rupture,  sliding,  segmental  ring, 

60, 

A,  E, 

265. 

Joint    of  rupture,    rotation,    segmental 

arch  surcharged  horizontally. 

74, 

no,  12 

t. 

271, 

348, 

340. 

Joint  of  rui>ture,  sliding,  segmental  arch 

."urcharged  horizontally. 

74. 

271 

Joint   of    rupture,    rotation,   segmental 

arch,  with  elevatid  ridge,  . 

80, 

117. 

276, 

344. 

Joint  of  rupture,  sliding,  segmental  arch, 

with  elevated  ridge. 

80. 

• 

276. 

Limit  thickness  of  pier,  circular  ring, 

43, 

46. 

230, 

232. 

"             "                 "     in  all  cases,    . 

64. 

256. 

"             "         of  arch,  circular  ring, 

47. 

233. 

INDEX. 


437 


Limit  iliickness  of  possible  eircu.  arches, 

surciiariccd  liurizoiitallv, 
Limit  thickness  of   possible    segmental 

arches  surcharged  horizontally, 
Limit    thickness  of  practicable  circiilar 

arches  surcharged  horizontally, 
Limit  thickness  of  practicable  segmental 

arches  surcharged  horizontallj'. 
Limit  thickness  of  arches  as  affected  by 

surcharge,    ..... 


Magazine  arch,  with  elevated  ridge — 
thrust,  ultimate,  by  rotation, 
"        actual,  " 

"  "  by  sliding,     . 

"         roof  inclined  45°,     . 
"  "    horizontal,    . 

thickness  of  pier,     .         .         .        •. 
"  "     examples, 

Monocacy  Bridge,       .... 
Mortar — common — no    element   of  sta- 

bility,  .         .         .         .         . 

Mortar — hydraulic — may    add    to    sta- 

ijiiity,   ' 

Mortar — effect  of,    . 

(See  thrust  of  circular,  magazine,  etc., 
arches. ) 


Pressure  per  unit  of  surface,    . 
"  "  "  limit, 

"        relative  at  the  key  and  at  the 
reins    ...... 

Pressure  on  the  joints  of  the  pier, 

"         determined  geometrically,   on 
the  several  joints. 
Point  of  application  of  the  thrust, 

Pier — thickness  of — general  conditions, 

"  "  "         formula?, 

Pier — thickness  of — limit,  . 

(See  circular,  elliptical,  etc,  arches.) 


Rupture  of  masonry  by  compression, 
"        joint  ot  (see  joint). 
"        third  mode,  the  key  rising,  . 


Segmental  ring  without  load, 

thickness  of  pier,  ... 

geometrical  method, 

Segmental    arches  surcharged    horizon- 
tally- 
ultimate  rotation  thrust,  etc.. 


Paragraiih  ami  Table. 

57,  142. 

U3. 

14-1. 

145. 

146. 


48—51,  F. 
117,  FF. 
52—54.  F,  FF. 
55,  56,  C,  F,  FF. 
57,  115,  116,  118,  D, 

Dl),  DDD. 
64—67,  128—135. 
67,  135. 
73,  136. 


15,  16,  17. 


112. 

136. 


140. 
141. 


153—159. 
4,6,23,111,  148,149. 

11,  12,  17,  I'J. 
64,  128. 


64. 


160. 

25,  142—146. 


Pasre. 


243,381. 

383. 

386. 

388. 

390. 


235—239,  320-1. 

344. 

239—341. 

241.  242. 

243,  335—343,  346. 

253—264,  359—371. 
260,  365. 
269,  373, 

200. 

200. 
205—210. 


329. 
372. 

379. 
379. 

404. 

2(12.    202,  213,  328, 

392—396. 
204,  210,  -211. 
253,  359. 

256. 


419. 

213,  380-390. 


68—73,  156,  157,  E.   ,  265—271,416. 
72,  73.  I  267—271. 

156,  157.  416. 


74—79,  E'. 


43S 


INDEX. 


nctunl  ri>tatii»n  tlinist,  i-tc, 

tlik-kne!*  of  j>i«'r.    .         .  .7'.' 

"  arch,  limit,       .         .  i  143,  145. 

Segmental  arches  «  itli  elevated  ridge,     I  8ii — 82. 
'■  "       a|>|>r«).\iiiiateforimilR\    85 — 89. 

Soarp  walls  stability  <>f,  .         .  [83,  84. 

Sureliarge,  its  effect  uii  the  ultimate  ro- 

tatioirthrusl, 13,  14,  IT 

Siircharire,  itd  effect  on  the  practicabil- 
ity ot  arches,        .         .  .         .  ,  146. 

Surcliar^rc  its  effect  on   the  actual  slid-  I 
iug  thrust,       .....        21 


Paragraph  and  Table. 
119,  12(>.  158,  EE. 


Thrust,  ultimate,  by  rotation, 

"  "  "         objections 

to  the  theory.  .... 

Thrust,  ultimate,  b\'  rotation,    general 

formuhe,       .... 
Thrust,  actual,  by  rotation. 

"  "  "  dependent  on. 

the  ]>ier, 

Thrust,  actual,  by  mtation,  of  the  well- 

estalilished  arcii,     .... 
Thrust,  actual,  by  sliding,  general  for- 

mulie,  ..... 

Thrust,  actual,  by  sliding, 

true,  generally  due  to  rotation, 
Tables  (see  table  of  contents). 
Thickness  of  pier  (see  pier). 

arch 

"  "     increase  of,  at  and  be- 

low the  rein.s,  .... 


Ultimate  thrust  (see  thrust). 


1— li»7. 

108. 

17. 
108—162. 

log,  110. 

113. 

20,  21. 
I— MT. 
22. 

136,  151,  152. 

137,  138,  IS'.t. 


Page. 

:i4s,  :i4<i,  417. 

275. 

:h83 — 389. 

276 — -SO. 

286 — 290. 

280. 

205.  210. 

39:>. 

212. 

199.  323. 

324. 

210. 

324 — 123. 

325—328. 

331. 

212. 

212. 

272— S89,  401—404. 
375—378. 


I 


/,  /  ///  , ./  ■  ■>ieh,-,l  ;\  •  Ko,/,,,'  11,1  ■./!  ."  J>  ( • 


• 


I 


AA/; 


m 


r^ 


,oh.-rlr,Ho.,,,.\t<>'J'"l>i 


") 


r 


hull   ol-yirhri-l  ,i- l<'"((/r  \i',<--fi''l*'' 


i.' 


A 


I'l.ATK  XII  . 


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Form  L9-100m-9.'52(A3105)444 


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